Uncertainty Theory, 3rd ed., 2009; Liu B

Uncertainty Theory

Third Edition

Baoding LiuUncertainty Theory Laboratory

Department of Mathematical Sciences

Tsinghua University

Beijing 100084, China

[email protected]

http://orsc.edu.cn/liu

3rd Edition c© 2009 by UTLAB

Japanese Translation Version c© 2008 by WAP

2nd Edition c© 2007 by Springer-Verlag Berlin

Chinese Version c© 2005 by Tsinghua University Press

1st Edition c© 2004 by Springer-Verlag Berlin

Reference to this book should be made as follows:Liu B, Uncertainty Theory, 3rd ed., http://orsc.edu.cn/liu/ut.pdf


http://orsc.edu.cn/utlab

mailto:[email protected]


http://orsc.edu.cn/utlab

http://orsc.edu.cn/liu/ut.pdf

Contents

Preface ix

1 Probability Theory 11.1 Probability Space . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Probability Distribution . . . . . . . . . . . . . . . . . . . . . 71.4 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Identical Distribution . . . . . . . . . . . . . . . . . . . . . . 131.6 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.8 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.9 Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . . 261.10 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.11 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.12 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.13 Convergence Concepts . . . . . . . . . . . . . . . . . . . . . . 351.14 Conditional Probability . . . . . . . . . . . . . . . . . . . . . 40

2 Credibility Theory 452.1 Credibility Space . . . . . . . . . . . . . . . . . . . . . . . . . 452.2 Fuzzy Variables . . . . . . . . . . . . . . . . . . . . . . . . . 572.3 Membership Function . . . . . . . . . . . . . . . . . . . . . . 582.4 Credibility Distribution . . . . . . . . . . . . . . . . . . . . . 622.5 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.6 Identical Distribution . . . . . . . . . . . . . . . . . . . . . . 702.7 Extension Principle of Zadeh . . . . . . . . . . . . . . . . . . 722.8 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . 732.9 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.10 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.11 Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . . 902.12 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.13 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.14 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

vi Contents

2.15 Convergence Concepts . . . . . . . . . . . . . . . . . . . . . . 1042.16 Conditional Credibility . . . . . . . . . . . . . . . . . . . . . 108

3 Chance Theory 1153.1 Chance Space . . . . . . . . . . . . . . . . . . . . . . . . . . 1153.2 Hybrid Variables . . . . . . . . . . . . . . . . . . . . . . . . . 1243.3 Chance Distribution . . . . . . . . . . . . . . . . . . . . . . . 1303.4 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . 1323.5 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353.6 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.7 Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.8 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.9 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.10 Convergence Concepts . . . . . . . . . . . . . . . . . . . . . . 1433.11 Conditional Chance . . . . . . . . . . . . . . . . . . . . . . . 147

4 Uncertainty Theory 1534.1 Uncertainty Space . . . . . . . . . . . . . . . . . . . . . . . . 1534.2 Uncertain Variables . . . . . . . . . . . . . . . . . . . . . . . 1624.3 Identification Function . . . . . . . . . . . . . . . . . . . . . 1644.4 Uncertainty Distribution . . . . . . . . . . . . . . . . . . . . 1704.5 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 1734.6 Identical Distribution . . . . . . . . . . . . . . . . . . . . . . 1744.7 Operational Law . . . . . . . . . . . . . . . . . . . . . . . . . 1754.8 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . 1764.9 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1834.10 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1854.11 Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . . 1874.12 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.13 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1904.14 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1914.15 Convergence Concepts . . . . . . . . . . . . . . . . . . . . . . 1944.16 Conditional Uncertainty . . . . . . . . . . . . . . . . . . . . . 195

5 Uncertain Process 2015.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 2015.2 Continuity Concepts . . . . . . . . . . . . . . . . . . . . . . . 2025.3 Renewal Process . . . . . . . . . . . . . . . . . . . . . . . . . 2025.4 Stationary Process . . . . . . . . . . . . . . . . . . . . . . . . 204

6 Uncertain Calculus 2056.1 Canonical Process . . . . . . . . . . . . . . . . . . . . . . . . 2056.2 Uncertain Integral . . . . . . . . . . . . . . . . . . . . . . . . 2076.3 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2086.4 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . 210

Contents vii

7 Uncertain Differential Equation 2117.1 Uncertain Differential Equation . . . . . . . . . . . . . . . . 2117.2 Existence and Uniqueness Theorem . . . . . . . . . . . . . . 2127.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2127.4 Uncertain Finance . . . . . . . . . . . . . . . . . . . . . . . . 2137.5 Uncertain Control . . . . . . . . . . . . . . . . . . . . . . . . 2167.6 Uncertain Filtering . . . . . . . . . . . . . . . . . . . . . . . 2167.7 Uncertain Dynamical System . . . . . . . . . . . . . . . . . . 217

8 Uncertain Logic 2198.1 Uncertain Proposition . . . . . . . . . . . . . . . . . . . . . . 2198.2 Uncertain Formula . . . . . . . . . . . . . . . . . . . . . . . . 2208.3 Truth Value . . . . . . . . . . . . . . . . . . . . . . . . . . . 2208.4 Some Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

9 Uncertain Inference 2259.1 Uncertain Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 2259.2 Inference Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 2269.3 Inference Rule with Multiple Antecedents . . . . . . . . . . . 2279.4 Inference Rule with Multiple If-Then Rules . . . . . . . . . . 2279.5 General Inference Rule . . . . . . . . . . . . . . . . . . . . . 228

A Measure and Integral 229A.1 Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . . . 229A.2 Classical Measures . . . . . . . . . . . . . . . . . . . . . . . . 231A.3 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . 232A.4 Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . 234

B Euler-Lagrange Equation 238

C Logic 239C.1 Traditional Logic . . . . . . . . . . . . . . . . . . . . . . . . . 239C.2 Probabilistic Logic . . . . . . . . . . . . . . . . . . . . . . . . 240C.3 Credibilistic Logic . . . . . . . . . . . . . . . . . . . . . . . . 240C.4 Hybrid Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 241C.5 Uncertain Logic . . . . . . . . . . . . . . . . . . . . . . . . . 243

D Inference 244D.1 Traditional Inference . . . . . . . . . . . . . . . . . . . . . . 244D.2 Fuzzy Inference . . . . . . . . . . . . . . . . . . . . . . . . . 244D.3 Hybrid Inference . . . . . . . . . . . . . . . . . . . . . . . . . 245D.4 Uncertain Inference . . . . . . . . . . . . . . . . . . . . . . . 246

E Maximum Uncertainty Principle 247

F Why Uncertain Measure? 248

viii Contents

Bibliography 251

List of Frequently Used Symbols 267

Index 268

Preface

There are various types of uncertainty. This fact provides a motivation tostudy the behavior of uncertain phenomena because “uncertainty is absoluteand certainty is relative” in the real world.

Randomness is a basic type of objective uncertainty, and probability the-ory is a branch of mathematics for studying the behavior of random phe-nomena. The study of probability theory was started by Pascal and Fermat(1654), and an axiomatic foundation of probability theory was given by Kol-mogoroff (1933) in his Foundations of Probability Theory. Probability theoryhas been widely applied in science and engineering. Chapter 1 will providethe probability theory.

Fuzziness is a basic type of subjective uncertainty initiated by Zadeh(1965). Credibility theory is a branch of mathematics for studying the be-havior of fuzzy phenomena. The study of credibility theory was started byLiu and Liu (2002), and an axiomatic foundation of credibility theory wasgiven by Liu (2004) in his Uncertainty Theory. Chapter 2 will introduce thecredibility theory.

Sometimes, fuzziness and randomness simultaneously appear in a sys-tem. A hybrid variable was proposed by Liu (2006) as a tool to describe thequantities with fuzziness and randomness. Both fuzzy random variable andrandom fuzzy variable are instances of hybrid variable. In addition, Li andLiu (2009) introduced the concept of chance measure for hybrid events. Afterthat, chance theory was developed steadily. Essentially, chance theory is ahybrid of probability theory and credibility theory. Chapter 3 will offer thechance theory.

A lot of surveys showed that the measure of union of events is not neces-sarily the maximum measure except those events are independent. This factimplies that many subjectively uncertain systems do not behave fuzzinessand membership function is no longer helpful. In order to deal with this typeof subjective uncertainty, Liu (2007) founded an uncertainty theory in hisUncertainty Theory, and had it become a branch of mathematics based onnormality, monotonicity, self-duality, countable subadditivity, and productmeasure axioms. Chapter 4 is devoted to the uncertainty theory.

For this new edition the entire text has been totally rewritten. More im-portantly, uncertain process, uncertain calculus, uncertain differential equa-tion, uncertain logic and uncertain inference are completely new. In addi-tion, uncertain finance, uncertain control, uncertain filtering and uncertaindynamical system have been added.

x Preface

The book is suitable for mathematicians, researchers, engineers, design-ers, and students in the field of mathematics, information science, operationsresearch, industrial engineering, computer science, artificial intelligence, fi-nance, and management science. The readers will learn the axiomatic ap-proach of uncertainty theory, and find this work a stimulating and usefulreference.

A Guide for the Reader

The logic dependence of chapters is illustrated by the figure below. Thereaders are not required to read the book from cover to cover. In fact, thereaders may skip over the first three chapters and start at Chapter 4.

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Acknowledgment

This work was supported by a series of grants from National Natural ScienceFoundation, Ministry of Education, and Ministry of Science and Technologyof China.

Baoding LiuTsinghua Universityhttp://orsc.edu.cn/liu

January 17, 2009


Chapter 1

Probability Theory

Probability measure is essentially a set function (i.e., a function whose ar-gument is a set) satisfying normality, nonnegativity and countable additivityaxioms. Probability theory is a branch of mathematics for studying the be-havior of random phenomena. The emphasis in this chapter is mainly onprobability space, random variable, probability distribution, independence,identical distribution, expected value, variance, moments, critical values, en-tropy, distance, convergence almost surely, convergence in probability, con-vergence in mean, convergence in distribution, and conditional probability.The main results in this chapter are well-known. For this reason the creditreferences are not provided.

1.1 Probability Space

Let Ω be a nonempty set, and A a σ-algebra over Ω. If Ω is countable, usuallyA is the power set of Ω. If Ω is uncountable, for example Ω = [0, 1], usuallyA is the Borel algebra of Ω. Each element in A is called an event.

In order to present an axiomatic definition of probability, it is necessaryto assign to each event A a number PrA which indicates the probabilitythat A will occur. In order to ensure that the number PrA has certainmathematical properties which we intuitively expect a probability to have,the following three axioms must be satisfied:

Axiom 1. (Normality) PrΩ = 1.

Axiom 2. (Nonnegativity) PrA ≥ 0 for any event A.

Axiom 3. (Countable Additivity) For every countable sequence of mutuallydisjoint events Ai, we have

Pr

∞⋃i=1

Ai

=

∞∑i=1

PrAi. (1.1)

2 Chapter 1 - Probability Theory

Definition 1.1 The set function Pr is called a probability measure if it sat-isfies the normality, nonnegativity, and countable additivity axioms.

Example 1.1: Let Ω = ω1, ω2, · · · , and let A be the power set of Ω.Assume that p1, p2, · · · are nonnegative numbers such that p1 + p2 + · · · = 1.Define a set function on A as

PrA =∑

ωi∈A

pi, A ∈ A. (1.2)

Then Pr is a probability measure.

Example 1.2: Let Ω = [0, 1] and let A be the Borel algebra over Ω. If Pr isthe Lebesgue measure, then Pr is a probability measure.

Example 1.3: Let φ be a nonnegative and integrable function on < (the setof real numbers) such that ∫

<φ(x)dx = 1. (1.3)

Then for any Borel set A, the set function

PrA =∫

A

φ(x)dx (1.4)

is a probability measure on <.

Theorem 1.1 Let Ω be a nonempty set, A a σ-algebra over Ω, and Pr aprobability measure. Then we have(a) Pr∅ = 0;(b) Pr is self-dual, i.e., PrA+ PrAc = 1 for any A ∈ A;(c) Pr is increasing, i.e., PrA ≤ PrB whenever A ⊂ B.

Proof: (a) Since ∅ and Ω are disjoint events and ∅ ∪ Ω = Ω, we havePr∅+ PrΩ = PrΩ which makes Pr∅ = 0.

(b) Since A and Ac are disjoint events and A∪Ac = Ω, we have PrA+PrAc = PrΩ = 1.

(c) Since A ⊂ B, we have B = A ∪ (B ∩ Ac), where A and B ∩ Ac aredisjoint events. Therefore PrB = PrA+ PrB ∩Ac ≥ PrA.

Probability Continuity Theorem

Theorem 1.2 (Probability Continuity Theorem) Let Ω be a nonempty set,A a σ-algebra over Ω, and Pr a probability measure. If A1, A2, · · · ∈ A andlimi→∞Ai exists, then

limi→∞

PrAi = Pr

limi→∞

Ai

. (1.5)

Section 1.1 - Probability Space 3

Proof: Step 1: Suppose Ai is an increasing sequence. Write Ai → A andA0 = ∅. Then Ai\Ai−1 is a sequence of disjoint events and

∞⋃i=1

(Ai\Ai−1) = A,k⋃

i=1

(Ai\Ai−1) = Ak

for k = 1, 2, · · · Thus we have

PrA = Pr ∞⋃

i=1

(Ai\Ai−1)

=∞∑

i=1

Pr Ai\Ai−1

= limk→∞

k∑i=1

Pr Ai\Ai−1 = limk→∞

Pr

k⋃i=1

(Ai\Ai−1)

= limk→∞

PrAk.

Step 2: If Ai is a decreasing sequence, then the sequence A1\Ai isclearly increasing. It follows that

PrA1 − PrA = Pr

limi→∞

(A1\Ai)

= limi→∞

Pr A1\Ai

= PrA1 − limi→∞

PrAi

which implies that PrAi → PrA.Step 3: If Ai is a sequence of events such that Ai → A, then for each

k, we have∞⋂

i=k

Ai ⊂ Ak ⊂∞⋃

i=k

Ai.

Since Pr is increasing, we have

Pr

∞⋂i=k

Ai

≤ PrAk ≤ Pr

∞⋃i=k

Ai

.

Note that∞⋂

i=k

Ai ↑ A,∞⋃

i=k

Ai ↓ A.

It follows from Steps 1 and 2 that PrAi → PrA.

Probability Space

Definition 1.2 Let Ω be a nonempty set, A a σ-algebra over Ω, and Pr aprobability measure. Then the triplet (Ω,A,Pr) is called a probability space.


Example 1.4: Let Ω = ω1, ω2, · · · , A the power set of Ω, and Pr aprobability measure defined by (1.2). Then (Ω,A,Pr) is a probability space.

Example 1.5: Let Ω = [0, 1], A the Borel algebra over Ω, and Pr theLebesgue measure. Then ([0, 1],A,Pr) is a probability space, and sometimesis called Lebesgue unit interval. For many purposes it is sufficient to use itas the basic probability space.

Product Probability Space

Let (Ωi,Ai,Pri), i = 1, 2, · · · , n be probability spaces, and Ω = Ω1 × Ω2 ×· · · × Ωn, A = A1 × A2 × · · · × An. Note that the probability measuresPri, i = 1, 2, · · · , n are finite. It follows from the product measure theoremthat there is a unique measure Pr on A such that

PrA1 ×A2 × · · · ×An = Pr1A1 × Pr2A2 × · · · × PrnAn

for any Ai ∈ Ai, i = 1, 2, · · · , n. This conclusion is called the product proba-bility theorem. The measure Pr is also a probability measure since

PrΩ = Pr1Ω1 × Pr2Ω2 × · · · × PrnΩn = 1.

Such a probability measure is called the product probability measure, denotedby Pr = Pr1 × Pr2 × · · · × Prn.

Definition 1.3 Let (Ωi,Ai,Pri), i = 1, 2, · · · , n be probability spaces, andΩ = Ω1×Ω2×· · ·×Ωn, A = A1×A2×· · ·×An, Pr = Pr1×Pr2× · · ·×Prn.Then the triplet (Ω,A,Pr) is called the product probability space.

Infinite Product Probability Space

Let (Ωi,Ai,Pri), i = 1, 2, · · · be an arbitrary sequence of probability spaces,and

Ω = Ω1 × Ω2 × · · · , A = A1 ×A2 × · · · (1.6)

It follows from the infinite product measure theorem that there is a uniqueprobability measure Pr on A such that

Pr A1 × · · · ×An × Ωn+1 × Ωn+2 × · · · = Pr1A1 × · · · × PrnAn

for any measurable rectangle A1 × · · · × An × Ωn+1 × Ωn+2 × · · · and alln = 1, 2, · · · The probability measure Pr is called the infinite product ofPri, i = 1, 2, · · · and is denoted by

Pr = Pr1 × Pr2 × · · · (1.7)

Definition 1.4 Let (Ωi,Ai,Pri), i = 1, 2, · · · be probability spaces, and Ω =Ω1 × Ω2 × · · · , A = A1 × A2 × · · · , Pr = Pr1×Pr2× · · · Then the triplet(Ω,A,Pr) is called the infinite product probability space.

Section 1.2 - Random Variables 5

1.2 Random Variables

Definition 1.5 A random variable is a measurable function from a proba-bility space (Ω,A,Pr) to the set of real numbers, i.e., for any Borel set B ofreal numbers, the set

ξ ∈ B = ω ∈ Ω∣∣ ξ(ω) ∈ B (1.8)

is an event.

Example 1.6: Take (Ω,A,Pr) to be ω1, ω2 with Prω1 = Prω2 = 0.5.Then the function

ξ(ω) =

0, if ω = ω1

1, if ω = ω2

is a random variable.

Example 1.7: Take (Ω,A,Pr) to be the interval [0, 1] with Borel algebraand Lebesgue measure. We define ξ as an identity function from Ω to [0,1].Since ξ is a measurable function, it is a random variable.

Example 1.8: A deterministic number c may be regarded as a special ran-dom variable. In fact, it is the constant function ξ(ω) ≡ c on the probabilityspace (Ω,A,Pr).

Definition 1.6 A random variable ξ is said to be(a) nonnegative if Prξ < 0 = 0;(b) positive if Prξ ≤ 0 = 0;(c) continuous if Prξ = x = 0 for each x ∈ <;(d) simple if there exists a finite sequence x1, x2, · · · , xm such that

Pr ξ 6= x1, ξ 6= x2, · · · , ξ 6= xm = 0; (1.9)

(e) discrete if there exists a countable sequence x1, x2, · · · such that

Pr ξ 6= x1, ξ 6= x2, · · · = 0. (1.10)

Definition 1.7 Let ξ1 and ξ2 be random variables defined on the probabilityspace (Ω,A,Pr). We say ξ1 = ξ2 if ξ1(ω) = ξ2(ω) for almost all ω ∈ Ω.

Random Vector

Definition 1.8 An n-dimensional random vector is a measurable functionfrom a probability space (Ω,A,Pr) to the set of n-dimensional real vectors,i.e., for any Borel set B of <n, the set

ξ ∈ B =ω ∈ Ω

∣∣ ξ(ω) ∈ B

(1.11)

is an event.


Theorem 1.3 The vector (ξ1, ξ2, · · · , ξn) is a random vector if and only ifξ1, ξ2, · · · , ξn are random variables.

Proof: Write ξ = (ξ1, ξ2, · · · , ξn). Suppose that ξ is a random vector on theprobability space (Ω,A,Pr). For any Borel set B of <, the set B × <n−1 isalso a Borel set of <n. Thus we have

ξ1 ∈ B = ξ1 ∈ B, ξ2 ∈ <, · · · , ξn ∈ < = ξ ∈ B ×<n−1 ∈ A

which implies that ξ1 is a random variable. A similar process may prove thatξ2, ξ3, · · · , ξn are random variables.

Conversely, suppose that all ξ1, ξ2, · · · , ξn are random variables on theprobability space (Ω,A,Pr). We define

B =B ⊂ <n

∣∣ ξ ∈ B ∈ A .The vector ξ = (ξ1, ξ2, · · · , ξn) is proved to be a random vector if we canprove that B contains all Borel sets of <n. First, the class B contains allopen intervals of <n because

ξ ∈n∏

i=1

(ai, bi)

=

n⋂i=1

ξi ∈ (ai, bi) ∈ A.

Next, the class B is a σ-algebra of <n because (i) we have <n ∈ B sinceξ ∈ <n = Ω ∈ A; (ii) if B ∈ B, then ξ ∈ B ∈ A, and

ξ ∈ Bc = ξ ∈ Bc ∈ A

which implies that Bc ∈ B; (iii) if Bi ∈ B for i = 1, 2, · · · , then ξ ∈ Bi ∈ Aand

ξ ∈∞⋃

i=1

Bi

=

∞⋃i=1

ξ ∈ Bi ∈ A

which implies that ∪iBi ∈ B. Since the smallest σ-algebra containing allopen intervals of <n is just the Borel algebra of <n, the class B contains allBorel sets of <n. The theorem is proved.

Random Arithmetic

In this subsections, we will suppose that all random variables are defined on acommon probability space. Otherwise, we may embed them into the productprobability space.

Definition 1.9 Let f : <n → < be a measurable function, and ξ1, ξ2, · · · , ξnrandom variables defined on the probability space (Ω,A,Pr). Then ξ =f(ξ1, ξ2, · · · , ξn) is a random variable defined by

ξ(ω) = f(ξ1(ω), ξ2(ω), · · · , ξn(ω)), ∀ω ∈ Ω. (1.12)

Section 1.3 - Probability Distribution 7

Example 1.9: Let ξ1 and ξ2 be random variables on the probability space(Ω,A,Pr). Then their sum is

(ξ1 + ξ2)(ω) = ξ1(ω) + ξ2(ω), ∀ω ∈ Ω

and their product is

(ξ1 × ξ2)(ω) = ξ1(ω)× ξ2(ω), ∀ω ∈ Ω.

The reader may wonder whether ξ(ω1, ω2, · · · , ωn) defined by (1.9) is arandom variable. The following theorem answers this question.

Theorem 1.4 Let ξ be an n-dimensional random vector, and f : <n → < ameasurable function. Then f(ξ) is a random variable.

Proof: Assume that ξ is a random vector on the probability space (Ω,A,Pr).For any Borel set B of <, since f is a measurable function, f−1(B) is also aBorel set of <n. Thus we have

f(ξ) ∈ B =ξ ∈ f−1(B)

∈ A

which implies that f(ξ) is a random variable.

1.3 Probability Distribution

Definition 1.10 The probability distribution Φ: < → [0, 1] of a randomvariable ξ is defined by

Φ(x) = Pr ξ ≤ x . (1.13)

That is, Φ(x) is the probability that the random variable ξ takes a value lessthan or equal to x.

Example 1.10: Assume that the random variables ξ and η have the sameprobability distribution. One question is whether ξ = η or not. Gener-ally speaking, it is not true. Take (Ω,A,Pr) to be ω1, ω2 with Prω1 =Prω2 = 0.5. We now define two random variables as follows,

ξ(ω) =−1, if ω = ω1

1, if ω = ω2,η(ω) =

1, if ω = ω1

−1, if ω = ω2.

Then ξ and η have the same probability distribution,

Φ(x) =

0, if x < −1

0.5, if − 1 ≤ x < 11, if x ≥ 1.

However, it is clear that ξ 6= η in the sense of Definition 1.7.


Theorem 1.5 (Sufficient and Necessary Condition for Probability Distribu-tion) A function Φ : < → [0, 1] is a probability distribution if and only if it isan increasing and right-continuous function with

limx→−∞

Φ(x) = 0; limx→+∞

Φ(x) = 1. (1.14)

Proof: For any x, y ∈ < with x < y, we have

Φ(y)− Φ(x) = Prx < ξ ≤ y ≥ 0.

Thus the probability distribution Φ is increasing. Next, let εi be a sequenceof positive numbers such that εi → 0 as i → ∞. Then, for every i ≥ 1, wehave

Φ(x+ εi)− Φ(x) = Prx < ξ ≤ x+ εi.

It follows from the probability continuity theorem that

limi→∞

Φ(x+ εi)− Φ(x) = Pr∅ = 0.

Hence Φ is a right-continuous function. Finally,

limx→−∞

Φ(x) = limx→−∞

Prξ ≤ x = Pr∅ = 0,

limx→+∞

Φ(x) = limx→+∞

Prξ ≤ x = PrΩ = 1.

Conversely, it is known there is a unique probability measure Pr on the Borelalgebra over < such that Pr(−∞, x] = Φ(x) for all x ∈ <. Furthermore,it is easy to verify that the random variable defined by ξ(x) = x from theprobability space (<,A,Pr) to < has the probability distribution Φ.

Remark 1.1: Theorem 1.5 states that the identity function is a universalfunction for any probability distribution by defining an appropriate proba-bility space. In fact, there is a universal probability space for any probabilitydistribution by defining an appropriate function.

Theorem 1.6 Let Φ be a probability distribution. Then there is a randomvariable on Lebesgue unit interval ([0, 1],A,Pr) whose probability distributionis just Φ.

Proof: Define ξ(ω) = supx|Φ(x) ≤ ω on the Lebesgue unit interval. Thenξ is a random variable whose probability distribution is just Φ because

Prξ ≤ y = Prω∣∣ sup x|Φ(x) ≤ ω ≤ y

= Pr

ω∣∣ ω ≤ Φ(y)

= Φ(y)

for any y ∈ <.

Section 1.3 - Probability Distribution 9

Theorem 1.7 A random variable ξ with probability distribution Φ is(a) nonnegative if and only if Φ(x) = 0 for all x < 0;(b) positive if and only if Φ(x) = 0 for all x ≤ 0;(c) simple if and only if Φ is a simple function;(d) discrete if and only if Φ is a step function;(e) continuous if and only if Φ is a continuous function.

Proof: The parts (a), (b), (c) and (d) follow immediately from the definition.Next we prove the part (e). If ξ is a continuous random variable, thenPrξ = x = 0. It follows from the probability continuity theorem that

limy↑x

(Φ(x)− Φ(y)) = limy↑x

Pry < ξ ≤ x = Prξ = x = 0

which proves the left-continuity of Φ. Since a probability distribution isalways right-continuous, Φ is continuous. Conversely, if Φ is continuous,then we immediately have Prξ = x = 0 for each x ∈ <.

Definition 1.11 A continuous random variable is said to be (a) singular ifits probability distribution is a singular function; (b) absolutely continuous ifits probability distribution is an absolutely continuous function.

Probability Density Function

Definition 1.12 The probability density function φ: < → [0,+∞) of a ran-dom variable ξ is a function such that

Φ(x) =∫ x

−∞φ(y)dy (1.15)

holds for all x ∈ <, where Φ is the probability distribution of the randomvariable ξ.

Let φ : < → [0,+∞) be a measurable function such that∫ +∞−∞ φ(x)dx = 1.

Then φ is the probability density function of some random variable becauseΦ(x) =

∫ x

−∞ φ(y)dy is an increasing and continuous function satisfying (1.14).

Theorem 1.8 (Probability Inversion Theorem) Let ξ be a random variablewhose probability density function φ exists. Then for any Borel set B of <,we have

Prξ ∈ B =∫

B

φ(y)dy. (1.16)

Proof: Let C be the class of all subsets C of < for which the relation

Prξ ∈ C =∫

C

φ(y)dy (1.17)

holds. We will show that C contains all Borel sets of <. It follows from theprobability continuity theorem and relation (1.17) that C is a monotone class.


It is also clear that C contains all intervals of the form (−∞, a], (a, b], (b,∞)and ∅ since

Prξ ∈ (−∞, a] = Φ(a) =∫ a

−∞φ(y)dy,

Prξ ∈ (b,+∞) = Φ(+∞)− Φ(b) =∫ +∞

b

φ(y)dy,

Prξ ∈ (a, b] = Φ(b)− Φ(a) =∫ b

a

φ(y)dy,

Prξ ∈ ∅ = 0 =∫∅φ(y)dy

where Φ is the probability distribution of ξ. Let F be the algebra consisting ofall finite unions of disjoint sets of the form (−∞, a], (a, b], (b,∞) and ∅. Notethat for any disjoint sets C1, C2, · · · , Cm of F and C = C1 ∪ C2 ∪ · · · ∪ Cm,we have

Prξ ∈ C =m∑

j=1

Prξ ∈ Cj =m∑

j=1

∫Cj

φ(y)dy =∫

C

φ(y)dy.

That is, C ∈ C. Hence we have F ⊂ C. Since the smallest σ-algebra containingF is just the Borel algebra of <, the monotone class theorem implies that Ccontains all Borel sets of <.

Some Special Distributions

Uniform Distribution: A random variable ξ has a uniform distribution ifits probability density function is defined by

φ(x) =

1

b− a, if a ≤ x ≤ b

0, otherwise(1.18)

denoted by U(a, b), where a and b are given real numbers with a < b.

Exponential Distribution: A random variable ξ has an exponential dis-tribution if its probability density function is defined by

φ(x) =

1β

exp(−xβ

), if x ≥ 0

0, if x < 0(1.19)

denoted by EXP(β), where β is a positive number.

Normal Distribution: A random variable ξ has a normal distribution ifits probability density function is defined by

φ(x) =1

σ√

2πexp

(− (x− µ)2

2σ2

), x ∈ < (1.20)

denoted by N (µ, σ2), where µ and σ are real numbers.

Section 1.4 - Independence 11

Joint Probability Distribution

Definition 1.13 The joint probability distribution Φ : <n → [0, 1] of a ran-dom vector (ξ1, ξ2, · · · , ξn) is defined by

Φ(x1, x2, · · · , xn) = Pr ξ1 ≤ x1, ξ2 ≤ x2, · · · , ξn ≤ xn .

Definition 1.14 The joint probability density function φ: <n → [0,+∞) ofa random vector (ξ1, ξ2, · · · , ξn) is a function such that

Φ(x1, x2, · · · , xn) =∫ x1

−∞

∫ x2

−∞· · ·∫ xn

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn

holds for all (x1, x2, · · · , xn) ∈ <n, where Φ is the probability distribution ofthe random vector (ξ1, ξ2, · · · , ξn).

1.4 Independence

Definition 1.15 The random variables ξ1, ξ2, · · · , ξm are said to be indepen-dent if

Pr

m⋂

i=1

ξi ∈ Bi

=

m∏i=1

Prξi ∈ Bi (1.21)

for any Borel sets B1, B2, · · · , Bm of <.

Theorem 1.9 Let ξ1, ξ2, · · · , ξm be independent random variables, and f1,f2, · · · , fn are measurable functions. Then f1(ξ1), f2(ξ2), · · · , fm(ξm) are in-dependent random variables.

Proof: For any Borel sets of B1, B2, · · · , Bm of <, we have

Pr

m⋂

i=1

fi(ξi) ∈ Bi

= Pr

m⋂

i=1

ξi ∈ f−1i (Bi)

=m∏

i=1

Prξi ∈ f−1i (Bi) =

m∏i=1

Prfi(ξi) ∈ Bi.

Thus f1(ξ1), f2(ξ2), · · · , fm(ξm) are independent random variables.

Theorem 1.10 Let ξi be random variables with probability distributions Φi,i = 1, 2, · · · ,m, respectively, and Φ the probability distribution of the randomvector (ξ1, ξ2, · · · , ξm). Then ξ1, ξ2, · · · , ξm are independent if and only if

Φ(x1, x2, · · · , xm) = Φ1(x1)Φ2(x2) · · ·Φm(xm) (1.22)

for all (x1, x2, · · · , xm) ∈ <m.


Proof: If ξ1, ξ2, · · · , ξm are independent random variables, then we have

Φ(x1, x2, · · · , xm) = Prξ1 ≤ x1, ξ2 ≤ x2, · · · , ξm ≤ xm= Prξ1 ≤ x1Prξ2 ≤ x2 · · ·Prξm ≤ xm= Φ1(x1)Φ2(x2) · · ·Φm(xm)

for all (x1, x2, · · · , xm) ∈ <m.Conversely, assume that (1.22) holds. Let x2, x3, · · · , xm be fixed real

numbers, and C the class of all subsets C of < for which the relation

Prξ1 ∈ C, ξ2 ≤ x2, · · · , ξm ≤ xm = Prξ1 ∈ Cm∏

i=2

Prξi ≤ xi (1.23)

holds. We will show that C contains all Borel sets of <. It follows fromthe probability continuity theorem and relation (1.23) that C is a monotoneclass. It is also clear that C contains all intervals of the form (−∞, a], (a, b],(b,∞) and ∅. Let F be the algebra consisting of all finite unions of disjointsets of the form (−∞, a], (a, b], (b,∞) and ∅. Note that for any disjoint setsC1, C2, · · · , Ck of F and C = C1 ∪ C2 ∪ · · · ∪ Ck, we have

Prξ1 ∈ C, ξ2 ≤ x2, · · · , ξm ≤ xm

=m∑

j=1

Prξ1 ∈ Cj , ξ2 ≤ x2, · · · , ξm ≤ xm

= Prξ1 ∈ CPrξ2 ≤ x2 · · ·Prξm ≤ xm.


Applying the same reasoning to each ξi in turn, we obtain the indepen-dence of the random variables.

Theorem 1.11 Let ξi be random variables with probability density functionsφi, i = 1, 2, · · · ,m, respectively, and φ the probability density function of therandom vector (ξ1, ξ2, · · · , ξm). Then ξ1, ξ2, · · · , ξm are independent if andonly if

φ(x1, x2, · · · , xm) = φ1(x1)φ2(x2) · · ·φm(xm) (1.24)

for almost all (x1, x2, · · · , xm) ∈ <m.

Proof: If φ(x1, x2, · · · , xm) = φ1(x1)φ2(x2) · · ·φm(xm) a.e., then we have

Φ(x1, x2, · · · , xm) =∫ x1

−∞

∫ x2

−∞· · ·∫ xm

−∞φ(t1, t2, · · · , tm)dt1dt2 · · ·dtm

=∫ x1

−∞

∫ x2

−∞· · ·∫ xm

−∞φ1(t1)φ2(t2) · · ·φm(tm)dt1dt2 · · ·dtm

=∫ x1

−∞φ1(t1)dt1

∫ x2

−∞φ2(t2)dt2 · · ·

∫ xm

−∞φm(tm)dtm

= Φ1(x1)Φ2(x2) · · ·Φm(xm)

Section 1.5 - Identical Distribution 13

for all (x1, x2, · · · , xm) ∈ <m. Thus ξ1, ξ2, · · · , ξm are independent. Con-versely, if ξ1, ξ2, · · · , ξm are independent, then for any (x1, x2, · · · , xm) ∈ <m,we have Φ(x1, x2, · · · , xm) = Φ1(x1)Φ2(x2) · · ·Φm(xm). Hence

Φ(x1, x2, · · · , xm) =∫ x1

−∞

∫ x2

−∞· · ·∫ xm

−∞φ1(t1)φ2(t2) · · ·φm(tm)dt1dt2 · · ·dtm

which implies that φ(x1, x2, · · · , xm) = φ1(x1)φ2(x2) · · ·φm(xm) a.e.

Example 1.11: Let ξ1, ξ2, · · · , ξm be independent random variables withprobability density functions φ1, φ2, · · · , φm, respectively, and f : <m →< a measurable function. Then for any Borel set B of real numbers, theprobability Prf(ξ1, ξ2, · · · , ξm) ∈ B is∫ ∫

· · ·∫

f(x1,x2,··· ,xm)∈B

φ1(x1)φ2(x2) · · ·φm(xm)dx1dx2 · · ·dxm.

1.5 Identical Distribution

Definition 1.16 The random variables ξ and η are said to be identicallydistributed if

Prξ ∈ B = Prξ ∈ B (1.25)

for any Borel set B of <.

Theorem 1.12 The random variables ξ and η are identically distributed ifand only if they have the same probability distribution.

Proof: Let Φ and Ψ be the probability distributions of ξ and η, respectively.If ξ and η are identically distributed random variables, then, for any x ∈ <,we have

Φ(x) = Prξ ∈ (−∞, x] = Prη ∈ (−∞, x] = Ψ(x).

Thus ξ and η have the same probability distribution.Conversely, assume that ξ and η have the same probability distribution.

Let C be the class of all subsets C of < for which the relation

Prξ ∈ C = Prη ∈ C (1.26)

holds. We will show that C contains all Borel sets of <. It follows fromthe probability continuity theorem and relation (1.26) that C is a monotoneclass. It is also clear that C contains all intervals of the form (−∞, a], (a, b],(b,∞) and ∅ since ξ and η have the same probability distribution. Let F bethe algebra consisting of all finite unions of disjoint sets of the form (−∞, a],


(a, b], (b,∞) and ∅. Note that for any disjoint sets C1, C2, · · · , Ck of F andC = C1 ∪ C2 ∪ · · · ∪ Ck, we have

Prξ ∈ C =k∑

j=1

Prξ ∈ Cj =k∑

j=1

Prη ∈ Cj = Prη ∈ C.


Theorem 1.13 Let ξ and η be two random variables whose probability den-sity functions exist. Then ξ and η are identically distributed if and only ifthey have the same probability density function.

Proof: It follows from Theorem 1.12 that the random variables ξ and η areidentically distributed if and only if they have the same probability distribu-tion, if and only if they have the same probability density function.

1.6 Expected Value

Definition 1.17 Let ξ be a random variable. Then the expected value of ξis defined by

E[ξ] =∫ +∞

0

Prξ ≥ rdr −∫ 0

−∞Prξ ≤ rdr (1.27)

provided that at least one of the two integrals is finite.

Example 1.12: Let ξ be a uniformly distributed random variable U(a, b).If a ≥ 0, then Prξ ≤ r ≡ 0 when r ≤ 0, and

Prξ ≥ r =

1, if r ≤ a

(b− r)/(b− a), if a ≤ r ≤ b

0, if r ≥ b,

E[ξ] =

(∫ a

0

1dr +∫ b

a

b− r

b− adr +

∫ +∞

b

0dr

)−∫ 0

−∞0dr =

a+ b

2.

If b ≤ 0, then Prξ ≥ r ≡ 0 when r ≥ 0, and

Prξ ≤ r =

1, if r ≥ b

(r − a)/(b− a), if a ≤ r ≤ b

0, if r ≤ a,

Section 1.6 - Expected Value 15

E[ξ] =∫ +∞

0

0dr −

(∫ a

−∞0dr +

∫ b

a

r − a

b− adr +

∫ 0

b

1dr

)=a+ b

2.

If a < 0 < b, then

Prξ ≥ r =

(b− r)/(b− a), if 0 ≤ r ≤ b

0, if r ≥ b,

Prξ ≤ r =

0, if r ≤ a

(r − a)/(b− a), if a ≤ r ≤ 0,

E[ξ] =

(∫ b

0

b− r

b− adr +

∫ +∞

b

0dr

)−(∫ a

−∞0dr +

∫ 0

a

r − a

b− adr)

=a+ b

2.

Thus we always have the expected value (a+ b)/2.

Example 1.13: Let ξ be an exponentially distributed random variableEXP(β). Then its expected value is β.

Example 1.14: Let ξ be a normally distributed random variable N (µ, σ2).Then its expected value is µ.

Example 1.15: Assume that ξ is a discrete random variable taking valuesxi with probabilities pi, i = 1, 2, · · · ,m, respectively. It follows from thedefinition of expected value operator that

E[ξ] =m∑

i=1

pixi.

Theorem 1.14 Let ξ be a random variable whose probability density functionφ exists. If the Lebesgue integral

∫ +∞

−∞xφ(x)dx

is finite, then we have

E[ξ] =∫ +∞

−∞xφ(x)dx. (1.28)


Proof: It follows from Definition 1.17 and Fubini Theorem that

E[ξ] =∫ +∞

0

Prξ ≥ rdr −∫ 0

−∞Prξ ≤ rdr

=∫ +∞

0

[∫ +∞

r

φ(x)dx]

dr −∫ 0

−∞

[∫ r

−∞φ(x)dx

]dr

=∫ +∞

0

[∫ x

0

φ(x)dr]

dx−∫ 0

−∞

[∫ 0

x

φ(x)dr]

dx

=∫ +∞

0

xφ(x)dx+∫ 0

−∞xφ(x)dx

=∫ +∞

−∞xφ(x)dx.

The theorem is proved.

Theorem 1.15 Let ξ be a random variable with probability distribution Φ.If the Lebesgue-Stieltjes integral∫ +∞

−∞xdΦ(x)


E[ξ] =∫ +∞

−∞xdΦ(x). (1.29)

Proof: Since the Lebesgue-Stieltjes integral∫ +∞−∞ xdΦ(x) is finite, we imme-

diately have

limy→+∞

∫ y

0

xdΦ(x) =∫ +∞

0

xdΦ(x), limy→−∞

∫ 0

y

xdΦ(x) =∫ 0

−∞xdΦ(x)

and

limy→+∞

∫ +∞

y

xdΦ(x) = 0, limy→−∞

∫ y

−∞xdΦ(x) = 0.

It follows from∫ +∞

y

xdΦ(x) ≥ y

(lim

z→+∞Φ(z)− Φ(y)

)= y(1− Φ(y)) ≥ 0, if y > 0,

∫ y

−∞xdΦ(x) ≤ y

(Φ(y)− lim

z→−∞Φ(z)

)= yΦ(y) ≤ 0, if y < 0

thatlim

y→+∞y (1− Φ(y)) = 0, lim

y→−∞yΦ(y) = 0.


Let 0 = x0 < x1 < x2 < · · · < xn = y be a partition of [0, y]. Then we have

n−1∑i=0

xi (Φ(xi+1)− Φ(xi)) →∫ y

0

xdΦ(x)

andn−1∑i=0

(1− Φ(xi+1))(xi+1 − xi) →∫ y

0

Prξ ≥ rdr

as max|xi+1 − xi| : i = 0, 1, · · · , n− 1 → 0. Since

n−1∑i=0

xi (Φ(xi+1)− Φ(xi))−n−1∑i=0

(1− Φ(xi+1))(xi+1 − xi) = y(Φ(y)− 1) → 0

as y → +∞. This fact implies that∫ +∞

0

Prξ ≥ rdr =∫ +∞

0

xdΦ(x).

A similar way may prove that

−∫ 0

−∞Prξ ≤ rdr =

∫ 0

−∞xdΦ(x).

Thus (1.29) is verified by the above two equations.

Linearity of Expected Value Operator

Theorem 1.16 Let ξ and η be random variables with finite expected values.Then for any numbers a and b, we have

E[aξ + bη] = aE[ξ] + bE[η]. (1.30)

Proof: Step 1: We first prove that E[ξ + b] = E[ξ] + b for any real numberb. When b ≥ 0, we have

E[ξ + b] =∫ ∞

0

Prξ + b ≥ rdr −∫ 0

−∞Prξ + b ≤ rdr

=∫ ∞

0

Prξ ≥ r − bdr −∫ 0

−∞Prξ ≤ r − bdr

= E[ξ] +∫ b

0

(Prξ ≥ r − b+ Prξ < r − b) dr

= E[ξ] + b.

If b < 0, then we have

E[ξ + b] = E[ξ]−∫ 0

b

(Prξ ≥ r − b+ Prξ < r − b) dr = E[ξ] + b.


Step 2: We prove that E[aξ] = aE[ξ] for any real number a. If a = 0,then the equation E[aξ] = aE[ξ] holds trivially. If a > 0, we have

E[aξ] =∫ ∞

0

Praξ ≥ rdr −∫ 0

−∞Praξ ≤ rdr

=∫ ∞

0

Prξ ≥ r

a

dr −

∫ 0

−∞Prξ ≤ r

a

dr

= a

∫ ∞

0

Prξ ≥ r

a

d( ra

)− a

∫ 0

−∞Prξ ≤ r

a

d( ra

)= aE[ξ].

If a < 0, we have

E[aξ] =∫ ∞

0

Praξ ≥ rdr −∫ 0

−∞Praξ ≤ rdr

=∫ ∞

0

Prξ ≤ r

a

dr −

∫ 0

−∞Prξ ≥ r

a

dr

= a

∫ ∞

0

Prξ ≥ r

a

d( ra

)− a

∫ 0

−∞Prξ ≤ r

a

d( ra

)= aE[ξ].

Step 3: We prove that E[ξ + η] = E[ξ] + E[η] when both ξ and ηare nonnegative simple random variables taking values a1, a2, · · · , am andb1, b2, · · · , bn, respectively. Then ξ + η is also a nonnegative simple randomvariable taking values ai + bj , i = 1, 2, · · · ,m, j = 1, 2, · · · , n. Thus we have

E[ξ + η] =m∑

i=1

n∑j=1

(ai + bj) Prξ = ai, η = bj

=m∑

i=1

n∑j=1

ai Prξ = ai, η = bj+m∑

i=1

n∑j=1

bj Prξ = ai, η = bj

=m∑

i=1

ai Prξ = ai+n∑

j=1

bj Prη = bj

= E[ξ] + E[η].

Step 4: We prove that E[ξ + η] = E[ξ] + E[η] when both ξ and η arenonnegative random variables. For every i ≥ 1 and every ω ∈ Ω, we define

ξi(ω) =

k − 1

2i, if

k − 12i

≤ ξ(ω) <k

2i, k = 1, 2, · · · , i2i

i, if i ≤ ξ(ω),


ηi(ω) =

k − 1

2i, if

k − 12i

≤ η(ω) <k

2i, k = 1, 2, · · · , i2i

i, if i ≤ η(ω).

Then ξi, ηi and ξi + ηi are three sequences of nonnegative simplerandom variables such that ξi ↑ ξ, ηi ↑ η and ξi + ηi ↑ ξ + η as i→∞. Notethat the functions Prξi > r, Prηi > r, Prξi + ηi > r, i = 1, 2, · · · arealso simple. It follows from the probability continuity theorem that

Prξi > r ↑ Prξ > r, ∀r ≥ 0

as i→∞. Since the expected value E[ξ] exists, we have

E[ξi] =∫ +∞

0

Prξi > rdr →∫ +∞

0

Prξ > rdr = E[ξ]

as i→∞. Similarly, we may prove that E[ηi] → E[η] and E[ξi+ηi] → E[ξ+η]as i→∞. It follows from Step 3 that E[ξ + η] = E[ξ] + E[η].

Step 5: We prove that E[ξ+η] = E[ξ]+E[η] when ξ and η are arbitraryrandom variables. Define

ξi(ω) =

ξ(ω), if ξ(ω) ≥ −i−i, otherwise,

ηi(ω) =

η(ω), if η(ω) ≥ −i−i, otherwise.

Since the expected values E[ξ] and E[η] are finite, we have

limi→∞

E[ξi] = E[ξ], limi→∞

E[ηi] = E[η], limi→∞

E[ξi + ηi] = E[ξ + η].

Note that (ξi + i) and (ηi + i) are nonnegative random variables. It followsfrom Steps 1 and 4 that

E[ξ + η] = limi→∞

E[ξi + ηi]

= limi→∞

(E[(ξi + i) + (ηi + i)]− 2i)

= limi→∞

(E[ξi + i] + E[ηi + i]− 2i)

= limi→∞

(E[ξi] + i+ E[ηi] + i− 2i)

= limi→∞

E[ξi] + limi→∞

E[ηi]

= E[ξ] + E[η].

Step 6: The linearity E[aξ + bη] = aE[ξ] + bE[η] follows immediatelyfrom Steps 2 and 5. The theorem is proved.


Product of Independent Random Variables

Theorem 1.17 Let ξ and η be independent random variables with finite ex-pected values. Then the expected value of ξη exists and

E[ξη] = E[ξ]E[η]. (1.31)

Proof: Step 1: We first prove the case where both ξ and η are nonnega-tive simple random variables taking values a1, a2, · · · , am and b1, b2, · · · , bn,respectively. Then ξη is also a nonnegative simple random variable takingvalues aibj , i = 1, 2, · · · ,m, j = 1, 2, · · · , n. It follows from the independenceof ξ and η that

E[ξη] =m∑

i=1

n∑j=1

aibj Prξ = ai, η = bj

=m∑

i=1

n∑j=1

aibj Prξ = aiPrη = bj

=(

m∑i=1

ai Prξ = ai)(

n∑j=1

bj Prη = bj

)= E[ξ]E[η].

Step 2: Next we prove the case where ξ and η are nonnegative randomvariables. For every i ≥ 1 and every ω ∈ Ω, we define

ξi(ω) =

k − 1

2i, if

k − 12i

≤ ξ(ω) <k

2i, k = 1, 2, · · · , i2i

i, if i ≤ ξ(ω),

ηi(ω) =

k − 1

2i, if

k − 12i

≤ η(ω) <k

2i, k = 1, 2, · · · , i2i

i, if i ≤ η(ω).

Then ξi, ηi and ξiηi are three sequences of nonnegative simple randomvariables such that ξi ↑ ξ, ηi ↑ η and ξiηi ↑ ξη as i → ∞. It follows fromthe independence of ξ and η that ξi and ηi are independent. Hence we haveE[ξiηi] = E[ξi]E[ηi] for i = 1, 2, · · · It follows from the probability continuitytheorem that Prξi > r, i = 1, 2, · · · are simple functions such that

Prξi > r ↑ Prξ > r, for all r ≥ 0

as i→∞. Since the expected value E[ξ] exists, we have

E[ξi] =∫ +∞

0

Prξi > rdr →∫ +∞

0

Prξ > rdr = E[ξ]

as i → ∞. Similarly, we may prove that E[ηi] → E[η] and E[ξiηi] → E[ξη]as i→∞. Therefore E[ξη] = E[ξ]E[η].


Step 3: Finally, if ξ and η are arbitrary independent random variables,then the nonnegative random variables ξ+ and η+ are independent and soare ξ+ and η−, ξ− and η+, ξ− and η−. Thus we have

E[ξ+η+] = E[ξ+]E[η+], E[ξ+η−] = E[ξ+]E[η−],

E[ξ−η+] = E[ξ−]E[η+], E[ξ−η−] = E[ξ−]E[η−].

It follows thatE[ξη] = E[(ξ+ − ξ−)(η+ − η−)]

= E[ξ+η+]− E[ξ+η−]− E[ξ−η+] + E[ξ−η−]

= E[ξ+]E[η+]− E[ξ+]E[η−]− E[ξ−]E[η+] + E[ξ−]E[η−]

= (E[ξ+]− E[ξ−]) (E[η+]− E[η−])

= E[ξ+ − ξ−]E[η+ − η−]

= E[ξ]E[η]

which proves the theorem.

Expected Value of Function of Random Variable

Theorem 1.18 Let ξ be a random variable with probability distribution Φ,and f : < → < a measurable function. If the Lebesgue-Stieltjes integral∫ +∞

−∞f(x)dΦ(x)


E[f(ξ)] =∫ +∞

−∞f(x)dΦ(x). (1.32)

Proof: It follows from the definition of expected value operator that

E[f(ξ)] =∫ +∞

0

Prf(ξ) ≥ rdr −∫ 0

−∞Prf(ξ) ≤ rdr. (1.33)

If f is a nonnegative simple measurable function, i.e.,

f(x) =

a1, if x ∈ B1

a2, if x ∈ B2

· · ·am, if x ∈ Bm

where B1, B2, · · · , Bm are mutually disjoint Borel sets, then we have

E[f(ξ)] =∫ +∞

0

Prf(ξ) ≥ rdr =m∑

i=1

ai Prξ ∈ Bi

=m∑

i=1

ai

∫Bi

dΦ(x) =∫ +∞

−∞f(x)dΦ(x).


We next prove the case where f is a nonnegative measurable function. Letf1, f2, · · · be a sequence of nonnegative simple functions such that fi ↑ f asi→∞. We have proved that

E[fi(ξ)] =∫ +∞

0

Prfi(ξ) ≥ rdr =∫ +∞

−∞fi(x)dΦ(x).

In addition, it is also known that Prfi(ξ) > r ↑ Prf(ξ) > r as i→∞ forr ≥ 0. It follows from the monotone convergence theorem that

E[f(ξ)] =∫ +∞

0

Prf(ξ) > rdr

= limi→∞

∫ +∞

0

Prfi(ξ) > rdr

= limi→∞

∫ +∞

−∞fi(x)dΦ(x)

=∫ +∞

−∞f(x)dΦ(x).

Finally, if f is an arbitrary measurable function, then we have f = f+ − f−

andE[f(ξ)] = E[f+(ξ)− f−(ξ)] = E[f+(ξ)]− E[f−(ξ)]

=∫ +∞

−∞f+(x)dΦ(x)−

∫ +∞

−∞f−(x)dΦ(x)

=∫ +∞

−∞f(x)dΦ(x).


Sum of a Random Number of Random Variables

Theorem 1.19 (Wald Identity) Assume that ξi is a sequence of iid ran-dom variables, and η is a positive random integer (i.e., a random variabletaking “positive integer” values) that is independent of the sequence ξi.Then we have

E

[η∑

i=1

ξi

]= E[η]E[ξ1]. (1.34)

Proof: Since η is independent of the sequence ξi, we have

Pr

η∑

i=1

ξi ≥ r

=

∞∑k=1

Prη = kPr ξ1 + ξ2 + · · ·+ ξk ≥ r .

Section 1.7 - Variance 23

If ξi are nonnegative random variables, then we have

E

[η∑

i=1

ξi

]=∫ +∞

0

Pr

η∑

i=1

ξi ≥ r

dr

=∫ +∞

0

∞∑k=1

Prη = kPr ξ1 + ξ2 + · · ·+ ξk ≥ rdr

=∞∑

k=1

Prη = k∫ +∞

0

Pr ξ1 + ξ2 + · · ·+ ξk ≥ rdr

=∞∑

k=1

Prη = k (E[ξ1] + E[ξ2] + · · ·+ E[ξk])

=∞∑

k=1

Prη = kkE[ξ1] (by iid hypothesis)

= E[η]E[ξ1].

If ξi are arbitrary random variables, then ξi = ξ+i − ξ−i , and

E

[η∑

i=1

ξi

]= E

[η∑

i=1

(ξ+i − ξ−i )

]= E

[η∑

i=1

ξ+i −η∑

i=1

ξ−i

]

= E

[η∑

i=1

ξ+i

]− E

[η∑

i=1

ξ−i

]= E[η]E[ξ+1 ]− E[η]E[ξ−1 ]

= E[η](E[ξ+1 ]− E[ξ−1 ]) = E[η]E[ξ+1 − ξ−1 ] = E[η]E[ξ1].

The theorem is thus proved.

1.7 Variance

Definition 1.18 Let ξ be a random variable with finite expected value e.Then the variance of ξ is defined by V [ξ] = E[(ξ − e)2].

The variance of a random variable provides a measure of the spread of thedistribution around its expected value. A small value of variance indicatesthat the random variable is tightly concentrated around its expected value;and a large value of variance indicates that the random variable has a widespread around its expected value.

Example 1.16: Let ξ be a uniformly distributed random variable U(a, b).Then its expected value is (a+ b)/2 and variance is (b− a)2/12.

Example 1.17: Let ξ be an exponentially distributed random variableEXP(β). Then its expected value is β and variance is β2.


Example 1.18: Let ξ be a normally distributed random variable N (µ, σ2).Then its expected value is µ and variance is σ2.

Theorem 1.20 If ξ is a random variable whose variance exists, a and b arereal numbers, then V [aξ + b] = a2V [ξ].

Proof: It follows from the definition of variance that

V [aξ + b] = E[(aξ + b− aE[ξ]− b)2

]= a2E[(ξ − E[ξ])2] = a2V [ξ].

Theorem 1.21 Let ξ be a random variable with expected value e. ThenV [ξ] = 0 if and only if Prξ = e = 1.

Proof: If V [ξ] = 0, then E[(ξ − e)2] = 0. Thus we have∫ +∞

0

Pr(ξ − e)2 ≥ rdr = 0

which implies Pr(ξ−e)2 ≥ r = 0 for any r > 0. Hence we have Pr(ξ−e)2 =0 = 1, i.e., Prξ = e = 1.

Conversely, if Prξ = e = 1, then we have Pr(ξ − e)2 = 0 = 1 andPr(ξ − e)2 ≥ r = 0 for any r > 0. Thus

V [ξ] =∫ +∞

0

Pr(ξ − e)2 ≥ rdr = 0.

Theorem 1.22 If ξ1, ξ2, · · · , ξn are independent random variables with finiteexpected values, then

V [ξ1 + ξ2 + · · ·+ ξn] = V [ξ1] + V [ξ2] + · · ·+ V [ξn]. (1.35)


V

[n∑

i=1

ξi

]= E

[(ξ1 + ξ2 + · · ·+ ξn − E[ξ1]− E[ξ2]− · · · − E[ξn])2

]=

n∑i=1

E[(ξi − E[ξi])2

]+ 2

n−1∑i=1

n∑j=i+1

E [(ξi − E[ξi])(ξj − E[ξj ])] .

Since ξ1, ξ2, · · · , ξn are independent, E [(ξi − E[ξi])(ξj − E[ξj ])] = 0 for alli, j with i 6= j. Thus (1.35) holds.

Maximum Variance Theorem

Let ξ be a random variable that takes values in [a, b], but whose probabilitydistribution is otherwise arbitrary. If its expected value is given, what is thepossible maximum variance? The maximum variance theorem will answerthis question, thus playing an important role in treating games against nature.

Section 1.8 - Moments 25

Theorem 1.23 (Edmundson-Madansky Inequality) Let f be a convex func-tion on [a, b], and ξ a random variable that takes values in [a, b] and hasexpected value e. Then

E[f(ξ)] ≤ b− e

b− af(a) +

e− a

b− af(b). (1.36)

Proof: For each ω ∈ Ω, we have a ≤ ξ(ω) ≤ b and

ξ(ω) =b− ξ(ω)b− a

a+ξ(ω)− a

b− ab.

It follows from the convexity of f that

f(ξ(ω)) ≤ b− ξ(ω)b− a

f(a) +ξ(ω)− a

b− af(b).

Taking expected values on both sides, we obtain (1.36).

Theorem 1.24 (Maximum Variance Theorem) Let ξ be a random variablethat takes values in [a, b] and has expected value e. Then

V [ξ] ≤ (e− a)(b− e) (1.37)

and equality holds if the random variable ξ is determined by

Prξ = x =

b− e

b− a, if x = a

e− a

b− a, if x = b.

(1.38)

Proof: It follows from Theorem 1.23 immediately by defining f(x) = (x−e)2.It is also easy to verify that the random variable determined by (1.38) hasvariance (e− a)(b− e). The theorem is proved.

1.8 Moments

Definition 1.19 Let ξ be a random variable, and k a positive number. Then(a) the expected value E[ξk] is called the kth moment;(b) the expected value E[|ξ|k] is called the kth absolute moment;(c) the expected value E[(ξ − E[ξ])k] is called the kth central moment;(d) the expected value E[|ξ−E[ξ]|k] is called the kth absolute central moment.

Note that the first central moment is always 0, the first moment is justthe expected value, and the second central moment is just the variance.

Theorem 1.25 Let ξ be a nonnegative random variable, and k a positivenumber. Then the k-th moment

E[ξk] = k

∫ +∞

0

rk−1 Prξ ≥ rdr. (1.39)


Proof: It follows from the nonnegativity of ξ that

E[ξk] =∫ ∞

0

Prξk ≥ xdx =∫ ∞

0

Prξ ≥ rdrk = k

∫ ∞

0

rk−1 Prξ ≥ rdr.


Theorem 1.26 Let ξ be a random variable that takes values in [a, b] and hasexpected value e. Then for any positive integer k, the kth absolute momentand kth absolute central moment satisfy the following inequalities,

E[|ξ|k] ≤ b− e

b− a|a|k +

e− a

b− a|b|k, (1.40)

E[|ξ − e|k] ≤ b− e

b− a(e− a)k +

e− a

b− a(b− e)k. (1.41)

Proof: It follows from Theorem 1.23 immediately by defining f(x) = |x|kand f(x) = |x− e|k.

1.9 Critical Values

Let ξ be a random variable. In order to measure it, we may use its expectedvalue. Alternately, we may employ α-optimistic value and α-pessimistic valueas a ranking measure.

Definition 1.20 Let ξ be a random variable, and α ∈ (0, 1]. Then

ξsup(α) = supr∣∣ Pr ξ ≥ r ≥ α

(1.42)

is called the α-optimistic value of ξ, and

ξinf(α) = infr∣∣ Pr ξ ≤ r ≥ α

(1.43)

is called the α-pessimistic value of ξ.

This means that the random variable ξ will reach upwards of the α-optimistic value ξsup(α) at least α of time, and will be below the α-pessimisticvalue ξinf(α) at least α of time. The optimistic value is also called percentile.

Theorem 1.27 Let ξ be a random variable. Then we have

Prξ ≥ ξsup(α) ≥ α, Prξ ≤ ξinf(α) ≥ α (1.44)

where ξsup(α) and ξinf(α) are the α-optimistic and α-pessimistic values of therandom variable ξ, respectively.

Section 1.9 - Critical Values 27

Proof: It follows from the definition of the optimistic value that there existsan increasing sequence ri such that Prξ ≥ ri ≥ α and ri ↑ ξsup(α) asi → ∞. Since ω|ξ(ω) ≥ ri ↓ ω|ξ(ω) ≥ ξsup(α), it follows from theprobability continuity theorem that

Prξ ≥ ξsup(α) = limi→∞

Prξ ≥ ri ≥ α.

The inequality Prξ ≤ ξinf(α) ≥ α may be proved similarly.

Example 1.19: Note that Prξ ≥ ξsup(α) > α and Prξ ≤ ξinf(α) > αmay hold. For example,

ξ =

0 with probability 0.41 with probability 0.6.

If α = 0.8, then ξsup(0.8) = 0 which makes Prξ ≥ ξsup(0.8) = 1 > 0.8. Inaddition, ξinf(0.8) = 1 and Prξ ≤ ξinf(0.8) = 1 > 0.8.

Theorem 1.28 Let ξ be a random variable. Then we have(a) ξinf(α) is an increasing and left-continuous function of α;(b) ξsup(α) is a decreasing and left-continuous function of α.

Proof: (a) It is easy to prove that ξinf(α) is an increasing function of α.Next, we prove the left-continuity of ξinf(α) with respect to α. Let αi bean arbitrary sequence of positive numbers such that αi ↑ α. Then ξinf(αi)is an increasing sequence. If the limitation is equal to ξinf(α), then the left-continuity is proved. Otherwise, there exists a number z∗ such that

limi→∞

ξinf(αi) < z∗ < ξinf(α).

Thus Prξ ≤ z∗ ≥ αi for each i. Letting i → ∞, we get Prξ ≤ z∗ ≥ α.Hence z∗ ≥ ξinf(α). A contradiction proves the left-continuity of ξinf(α) withrespect to α. The part (b) may be proved similarly.

Theorem 1.29 Let ξ be a random variable. Then we have(a) if α > 0.5, then ξinf(α) ≥ ξsup(α);(b) if α ≤ 0.5, then ξinf(α) ≤ ξsup(α).

Proof: Part (a): Write ξ(α) = (ξinf(α) + ξsup(α))/2. If ξinf(α) < ξsup(α),then we have

1 ≥ Prξ < ξ(α)+ Prξ > ξ(α) ≥ α+ α > 1.

A contradiction proves ξinf(α) ≥ ξsup(α). Part (b): Assume that ξinf(α) >ξsup(α). It follows from the definition of ξinf(α) that Prξ ≤ ξ(α) < α.Similarly, it follows from the definition of ξsup(α) that Prξ ≥ ξ(α) < α.Thus

1 ≤ Prξ ≤ ξ(α)+ Prξ ≥ ξ(α) < α+ α ≤ 1.

A contradiction proves ξinf(α) ≤ ξsup(α). The theorem is proved.


Theorem 1.30 Let ξ be a random variable. Then we have(a) if c ≥ 0, then (cξ)sup(α) = cξsup(α) and (cξ)inf(α) = cξinf(α);(b) if c < 0, then (cξ)sup(α) = cξinf(α) and (cξ)inf(α) = cξsup(α).

Proof: (a) If c = 0, then it is obviously valid. When c > 0, we have

(cξ)sup(α) = sup r | Prcξ ≥ r ≥ α= c sup r/c | Pr ξ ≥ r/c ≥ α= cξsup(α).

A similar way may prove that (cξ)inf(α) = cξinf(α).(b) In order to prove this part, it suffices to verify that (−ξ)sup(α) =

−ξinf(α) and (−ξ)inf(α) = −ξsup(α). In fact, for any α ∈ (0, 1], we have

(−ξ)sup(α) = supr | Pr−ξ ≥ r ≥ α= − inf−r | Prξ ≤ −r ≥ α= −ξinf(α).

Similarly, we may prove that (−ξ)inf(α) = −ξsup(α). The theorem is proved.

1.10 Entropy

Given a random variable, what is the degree of difficulty of predicting thespecified value that the random variable will take? In order to answer thisquestion, Shannon [219] defined a concept of entropy as a measure of uncer-tainty.

Entropy of Discrete Random Variables

Definition 1.21 Let ξ be a discrete random variable taking values xi withprobabilities pi, i = 1, 2, · · · , respectively. Then its entropy is defined by

H[ξ] = −∞∑

i=1

pi ln pi. (1.45)

It should be noticed that the entropy depends only on the number ofvalues and their probabilities and does not depend on the actual values thatthe random variable takes.

Theorem 1.31 Let ξ be a discrete random variable taking values xi withprobabilities pi, i = 1, 2, · · · , respectively. Then

H[ξ] ≥ 0 (1.46)

and equality holds if and only if there exists an index k such that pk = 1, i.e.,ξ is essentially a deterministic number.

Section 1.10 - Entropy 29

................................................................................................................................................................................................................................................................................................................................... ...............

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

......................

...............

0 1t

S(t)

.....................................................................................

..................................................................................................................................................................................................................................................................................................................

1/e

1/e

...................................

Figure 1.1: Function S(t) = −t ln t is concave

Proof: The nonnegativity is clear. In addition, H[ξ] = 0 if and only ifpi = 0 or 1 for each i. That is, there exists one and only one index k suchthat pk = 1. The theorem is proved.

This theorem states that the entropy of a discrete random variable reachesits minimum 0 when the random variable degenerates to a deterministic num-ber. In this case, there is no uncertainty.

Theorem 1.32 Let ξ be a simple random variable taking values xi with prob-abilities pi, i = 1, 2, · · · , n, respectively. Then

H[ξ] ≤ lnn (1.47)

and equality holds if and only if pi ≡ 1/n for all i = 1, 2, · · · , n.

Proof: Since the function S(t) is a concave function of t and p1 + p2 + · · ·+pn = 1, we have

−n∑

i=1

pi ln pi ≤ −n

(1n

n∑i=1

pi

)ln

(1n

n∑i=1

pi

)= lnn

which implies that H[ξ] ≤ lnn and equality holds if and only if p1 = p2 =· · · = pn, i.e., pi ≡ 1/n for all i = 1, 2, · · · , n.

This theorem states that the entropy of a simple random variable reachesits maximum lnn when all outcomes are equiprobable. In this case, there isno preference among all the values that the random variable will take.

Entropy of Absolutely Continuous Random Variables

Definition 1.22 Let ξ be a random variable with probability density functionφ. Then its entropy is defined by

H[ξ] = −∫ +∞

−∞φ(x) lnφ(x)dx. (1.48)


Example 1.20: Let ξ be a uniformly distributed random variable on [a, b].Then its entropy is H[ξ] = ln(b− a). This example shows that the entropy ofabsolutely continuous random variable may assume both positive and nega-tive values since ln(b− a) < 0 if b− a < 1; and ln(b− a) > 0 if b− a > 1.

Example 1.21: Let ξ be an exponentially distributed random variable withexpected value β. Then its entropy is H[ξ] = 1 + lnβ.

Example 1.22: Let ξ be a normally distributed random variable with ex-pected value e and variance σ2. Then its entropy is H[ξ] = 1/2 + ln

√2πσ.

Maximum Entropy Principle

Given some constraints, for example, expected value and variance, there areusually multiple compatible probability distributions. For this case, we wouldlike to select the distribution that maximizes the value of entropy and satisfiesthe prescribed constraints. This method is often referred to as the maximumentropy principle (Jaynes [68]).

Example 1.23: Let ξ be an absolutely continuous random variable on [a, b].The maximum entropy principle attempts to find the probability densityfunction φ(x) that maximizes the entropy

−∫ b

a

φ(x) lnφ(x)dx

subject to the natural constraint∫ b

aφ(x)dx = 1. The Lagrangian is

L = −∫ b

a

φ(x) lnφ(x)dx− λ

(∫ b

a

φ(x)dx− 1

).

It follows from the Euler-Lagrange equation that the maximum entropy prob-ability density function meets

lnφ(x) + 1 + λ = 0

and has the form φ(x) = exp(−1 − λ). Substituting it into the naturalconstraint, we get

φ∗(x) =1

b− a, a ≤ x ≤ b

which is just the uniformly distributed random variable, and the maximumentropy is H[ξ∗] = ln(b− a).

Example 1.24: Let ξ be an absolutely continuous random variable on [0,∞).Assume that the expected value of ξ is prescribed to be β. The maximumentropy probability density function φ(x) should maximize the entropy

−∫ +∞

0

φ(x) lnφ(x)dx


subject to the constraints∫ +∞

0

φ(x)dx = 1,∫ +∞

0

xφ(x)dx = β.

The Lagrangian is

L = −∫ ∞

0

φ(x) lnφ(x)dx−λ1

(∫ ∞

0

φ(x)dx− 1)−λ2

(∫ ∞

0

xφ(x)dx− β

).

The maximum entropy probability density function meets Euler-Lagrangeequation

lnφ(x) + 1 + λ1 + λ2x = 0

and has the form φ(x) = exp(−1 − λ1 − λ2x). Substituting it into theconstraints, we get

φ∗(x) =1β

exp(−xβ

), x ≥ 0

which is just the exponentially distributed random variable, and the maxi-mum entropy is H[ξ∗] = 1 + lnβ.

Example 1.25: Let ξ be an absolutely continuous random variable on(−∞,+∞). Assume that the expected value and variance of ξ are prescribedto be µ and σ2, respectively. The maximum entropy probability densityfunction φ(x) should maximize the entropy

−∫ +∞

−∞φ(x) lnφ(x)dx

subject to the constraints∫ +∞

−∞φ(x)dx = 1,

∫ +∞

−∞xφ(x)dx = µ,

∫ +∞

−∞(x− µ)2φ(x)dx = σ2.

The Lagrangian is

L = −∫ +∞

−∞φ(x) lnφ(x)dx− λ1

(∫ +∞

−∞φ(x)dx− 1

)

−λ2

(∫ +∞

−∞xφ(x)dx− µ

)− λ3

(∫ +∞

−∞(x− µ)2φ(x)dx− σ2

).

The maximum entropy probability density function meets Euler-Lagrangeequation

lnφ(x) + 1 + λ1 + λ2x+ λ3(x− µ)2 = 0


and has the form φ(x) = exp(−1 − λ1 − λ2x − λ3(x − µ)2). Substituting itinto the constraints, we get

φ∗(x) =1

σ√

2πexp

(− (x− µ)2

2σ2

), x ∈ <

which is just the normally distributed random variable, and the maximumentropy is H[ξ∗] = 1/2 + ln

√2πσ.

1.11 Distance

Distance is a powerful concept in many disciplines of science and engineering.This section introduces the distance between random variables.

Definition 1.23 The distance between random variables ξ and η is definedas

d(ξ, η) = E[|ξ − η|]. (1.49)

Theorem 1.33 Let ξ, η, τ be random variables, and let d(·, ·) be the distance.Then we have(a) (Nonnegativity) d(ξ, η) ≥ 0;(b) (Identification) d(ξ, η) = 0 if and only if ξ = η;(c) (Symmetry) d(ξ, η) = d(η, ξ);(d) (Triangle Inequality) d(ξ, η) ≤ d(ξ, τ) + d(η, τ).

Proof: The parts (a), (b) and (c) follow immediately from the definition.The part (d) is proved by the following relation,

E[|ξ − η|] ≤ E[|ξ − τ |+ |η − τ |] = E[|ξ − τ |] + E[|η − τ |].

1.12 Inequalities

It is well-known that there are several inequalities in probability theory, suchas Markov inequality, Chebyshev inequality, Holder’s inequality, Minkowskiinequality, and Jensen’s inequality. They play an important role in boththeory and applications.

Theorem 1.34 Let ξ be a random variable, and f a nonnegative measurablefunction. If f is even (i.e., f(x) = f(−x) for any x ∈ <) and increasing on[0,∞), then for any given number t > 0, we have

Pr|ξ| ≥ t ≤ E[f(ξ)]f(t)

. (1.50)

Section 1.12 - Inequalities 33

Proof: It is clear that Pr|ξ| ≥ f−1(r) is a monotone decreasing functionof r on [0,∞). It follows from the nonnegativity of f(ξ) that

E[f(ξ)] =∫ +∞

0

Prf(ξ) ≥ rdr

=∫ +∞

0

Pr|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

Pr|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

dr · Pr|ξ| ≥ f−1(f(t))

= f(t) · Pr|ξ| ≥ t

which proves the inequality.

Theorem 1.35 (Markov Inequality) Let ξ be a random variable. Then forany given numbers t > 0 and p > 0, we have

Pr|ξ| ≥ t ≤ E[|ξ|p]tp

. (1.51)

Proof: It is a special case of Theorem 1.34 when f(x) = |x|p.

Example 1.26: For any given positive number t, we define a random variableas follows,

ξ =

0 with probability 1/2t with probability 1/2.

Then Prξ ≥ t = 1/2 = E[|ξ|p]/tp.

Theorem 1.36 (Chebyshev Inequality) Let ξ be a random variable whosevariance V [ξ] exists. Then for any given number t > 0, we have

Pr |ξ − E[ξ]| ≥ t ≤ V [ξ]t2

. (1.52)

Proof: It is a special case of Theorem 1.34 when the random variable ξ isreplaced with ξ − E[ξ] and f(x) = x2.

Example 1.27: For any given positive number t, we define a random variableas follows,

ξ =

−t with probability 1/2t with probability 1/2.

Then Pr|ξ − E[ξ]| ≥ t = 1 = V [ξ]/t2.


Theorem 1.37 (Holder’s Inequality) Let p and q be two positive real num-bers with 1/p+1/q = 1, and let ξ and η be random variables with E[|ξ|p] <∞and E[|η|q] <∞. Then we have

E[|ξη|] ≤ p√E[|ξ|p] q

√E[|η|q]. (1.53)

Proof: The inequality holds trivially if at least one of ξ and η is zero a.s. Nowwe assume E[|ξ|p] > 0 and E[|η|q] > 0. It is easy to prove that the functionf(x, y) = p

√x q√y is a concave function on D = (x, y) : x ≥ 0, y ≥ 0. Thus

for any point (x0, y0) with x0 > 0 and y0 > 0, there exist two real numbersa and b such that

f(x, y)− f(x0, y0) ≤ a(x− x0) + b(y − y0), ∀(x, y) ∈ D.

Letting x0 = E[|ξ|p], y0 = E[|η|q], x = |ξ|p and y = |η|q, we have

f(|ξ|p, |η|q)− f(E[|ξ|p], E[|η|q]) ≤ a(|ξ|p − E[|ξ|p]) + b(|η|q − E[|η|q]).

Taking the expected values on both sides, we obtain

E[f(|ξ|p, |η|q)] ≤ f(E[|ξ|p], E[|η|q]).

Hence the inequality (1.53) holds.

Theorem 1.38 (Minkowski Inequality) Let p be a real number with p ≥ 1,and let ξ and η be random variables with E[|ξ|p] <∞ and E[|η|p] <∞. Thenwe have

p√E[|ξ + η|p] ≤ p

√E[|ξ|p] + p

√E[|η|p]. (1.54)

Proof: The inequality holds trivially if at least one of ξ and η is zero a.s. Nowwe assume E[|ξ|p] > 0 and E[|η|p] > 0. It is easy to prove that the functionf(x, y) = ( p

√x + p

√y)p is a concave function on D = (x, y) : x ≥ 0, y ≥ 0.

Thus for any point (x0, y0) with x0 > 0 and y0 > 0, there exist two realnumbers a and b such that


Letting x0 = E[|ξ|p], y0 = E[|η|p], x = |ξ|p and y = |η|p, we have

f(|ξ|p, |η|p)− f(E[|ξ|p], E[|η|p]) ≤ a(|ξ|p − E[|ξ|p]) + b(|η|p − E[|η|p]).


E[f(|ξ|p, |η|p)] ≤ f(E[|ξ|p], E[|η|p]).


Section 1.13 - Convergence Concepts 35

Theorem 1.39 (Jensen’s Inequality) Let ξ be a random variable, and f :< → < a convex function. If E[ξ] and E[f(ξ)] are finite, then

f(E[ξ]) ≤ E[f(ξ)]. (1.55)

Especially, when f(x) = |x|p and p ≥ 1, we have |E[ξ]|p ≤ E[|ξ|p].

Proof: Since f is a convex function, for each y, there exists a number k suchthat f(x)− f(y) ≥ k · (x− y). Replacing x with ξ and y with E[ξ], we obtain

f(ξ)− f(E[ξ]) ≥ k · (ξ − E[ξ]).

Taking the expected values on both sides, we have

E[f(ξ)]− f(E[ξ]) ≥ k · (E[ξ]− E[ξ]) = 0


1.13 Convergence Concepts

There are four main types of convergence concepts of random sequence:convergence almost surely (a.s.), convergence in probability, convergence inmean, and convergence in distribution.

Table 1.1: Relations among Convergence Concepts

Convergence Almost Surely Convergence → Convergence in Probability in Distribution

Convergence in Mean

Definition 1.24 Suppose that ξ, ξ1, ξ2, · · · are random variables defined onthe probability space (Ω,A,Pr). The sequence ξi is said to be convergenta.s. to ξ if and only if there exists a set A ∈ A with PrA = 1 such that

limi→∞

|ξi(ω)− ξ(ω)| = 0 (1.56)

for every ω ∈ A. In that case we write ξi → ξ, a.s.

Definition 1.25 Suppose that ξ, ξ1, ξ2, · · · are random variables defined onthe probability space (Ω,A,Pr). We say that the sequence ξi converges inprobability to ξ if

limi→∞

Pr |ξi − ξ| ≥ ε = 0 (1.57)

for every ε > 0.


Definition 1.26 Suppose that ξ, ξ1, ξ2, · · · are random variables with finiteexpected values on the probability space (Ω,A,Pr). We say that the sequenceξi converges in mean to ξ if

limi→∞

E[|ξi − ξ|] = 0. (1.58)

In addition, the sequence ξi is said to converge in mean square to ξ if

limi→∞

E[|ξi − ξ|2] = 0. (1.59)

Definition 1.27 Suppose that Φ,Φ1,Φ2, · · · are the probability distributionsof random variables ξ, ξ1, ξ2, · · · , respectively. We say that ξi converges indistribution to ξ if Φi → Φ at any continuity point of Φ.

Convergence Almost Surely vs. Convergence in Probability

Theorem 1.40 Suppose that ξ, ξ1, ξ2, · · · are random variables defined onthe probability space (Ω,A,Pr). Then ξi converges a.s. to ξ if and only if,for every ε > 0, we have

limn→∞

Pr

∞⋃i=n

|ξi − ξ| ≥ ε

= 0. (1.60)

Proof: For every i ≥ 1 and ε > 0, we define

X =ω ∈ Ω

∣∣ limi→∞

ξi(ω) 6= ξ(ω),

Xi(ε) =ω ∈ Ω

∣∣ |ξi(ω)− ξ(ω)| ≥ ε.

It is clear that

X =⋃ε>0

( ∞⋂n=1

∞⋃i=n

Xi(ε)

).

Note that ξi → ξ, a.s. if and only if PrX = 0. That is, ξi → ξ, a.s. if andonly if

Pr

∞⋂n=1

∞⋃i=n

Xi(ε)

= 0

for every ε > 0. Since∞⋃

i=n

Xi(ε) ↓∞⋂

n=1

∞⋃i=n

Xi(ε),

it follows from the probability continuity theorem that

limn→∞

Pr

∞⋃i=n

Xi(ε)

= Pr

∞⋂n=1

∞⋃i=n

Xi(ε)

= 0.



Theorem 1.41 Suppose that ξ, ξ1, ξ2, · · · are random variables defined onthe probability space (Ω,A,Pr). If ξi converges a.s. to ξ, then ξi convergesin probability to ξ.

Proof: It follows from the convergence a.s. and Theorem 1.40 that

limn→∞

Pr

∞⋃i=n

|ξi − ξ| ≥ ε

= 0

for each ε > 0. For every n ≥ 1, since

|ξn − ξ| ≥ ε ⊂∞⋃

i=n

|ξi − ξ| ≥ ε,

we have Pr|ξn − ξ| ≥ ε → 0 as n→∞. Hence the theorem holds.

Example 1.28: Convergence in probability does not imply convergence a.s.For example, take (Ω,A,Pr) to be the interval [0, 1] with Borel algebra andLebesgue measure. For any positive integer i, there is an integer j such thati = 2j + k, where k is an integer between 0 and 2j − 1. We define a randomvariable on Ω by

ξi(ω) =

1, if k/2j ≤ ω ≤ (k + 1)/2j

0, otherwise

for i = 1, 2, · · · and ξ = 0. For any small number ε > 0, we have

Pr |ξi − ξ| ≥ ε =12j→ 0

as i→∞. That is, the sequence ξi converges in probability to ξ. However,for any ω ∈ [0, 1], there is an infinite number of intervals of the form [k/2j , (k+1)/2j ] containing ω. Thus ξi(ω) 6→ 0 as i→∞. In other words, the sequenceξi does not converge a.s. to ξ.

Convergence in Probability vs. Convergence in Mean

Theorem 1.42 Suppose that ξ, ξ1, ξ2, · · · are random variables defined onthe probability space (Ω,A,Pr). If the sequence ξi converges in mean to ξ,then ξi converges in probability to ξ.

Proof: It follows from the Markov inequality that, for any given numberε > 0,

Pr |ξi − ξ| ≥ ε ≤ E[|ξi − ξ|]ε

→ 0

as i→∞. Thus ξi converges in probability to ξ.


Example 1.29: Convergence in probability does not imply convergence inmean. For example, take (Ω,A,Pr) to be ω1, ω2, · · · with Prωj = 1/2j

for j = 1, 2, · · · The random variables are defined by

ξiωj =

2i, if j = i0, otherwise


Pr |ξi − ξ| ≥ ε =12i→ 0.

That is, the sequence ξi converges in probability to ξ. However, we have

E [|ξi − ξ|] = 2i · 12i

= 1.

That is, the sequence ξi does not converge in mean to ξ.

Convergence Almost Surely vs. Convergence in Mean

Example 1.30: Convergence a.s. does not imply convergence in mean. Forexample, take (Ω,A,Pr) to be ω1, ω2, · · · with Prωj = 1/2j for j =1, 2, · · · The random variables are defined by

ξiωj =

2i, if j = i0, otherwise

for i = 1, 2, · · · and ξ = 0. Then ξi converges a.s. to ξ. However, thesequence ξi does not converge in mean to ξ.

Example 1.31: Convergence in mean does not imply convergence a.s. Forexample, take (Ω,A,Pr) to be the interval [0, 1] with Borel algebra andLebesgue measure. For any positive integer i, there is an integer j suchthat i = 2j + k, where k is an integer between 0 and 2j − 1. We define arandom variable on Ω by

ξi(ω) =

1, if k/2j ≤ ω ≤ (k + 1)/2j

0, otherwise

for i = 1, 2, · · · and ξ = 0. Then

E [|ξi − ξ|] =12j→ 0.

That is, the sequence ξi converges in mean to ξ. However, ξi does notconverge a.s. to ξ.

Section 1.14 - Conditional Probability 39

Convergence in Probability vs. Convergence in Distribution

Theorem 1.43 Suppose that ξ, ξ1, ξ2, · · · are random variables defined onthe probability space (Ω,A,Pr). If the sequence ξi converges in probabilityto ξ, then ξi converges in distribution to ξ.

Proof: Let x be any given continuity point of the distribution Φ. On theone hand, for any y > x, we have

ξi ≤ x = ξi ≤ x, ξ ≤ y ∪ ξi ≤ x, ξ > y ⊂ ξ ≤ y ∪ |ξi − ξ| ≥ y − x

which implies that

Φi(x) ≤ Φ(y) + Pr|ξi − ξ| ≥ y − x.

Since ξi converges in probability to ξ, we have Pr|ξi − ξ| ≥ y − x → 0.Thus we obtain lim supi→∞Φi(x) ≤ Φ(y) for any y > x. Letting y → x, weget

lim supi→∞

Φi(x) ≤ Φ(x). (1.61)

On the other hand, for any z < x, we have

ξ ≤ z = ξ ≤ z, ξi ≤ x ∪ ξ ≤ z, ξi > x ⊂ ξi ≤ x ∪ |ξi − ξ| ≥ x− z

which implies that

Φ(z) ≤ Φi(x) + Pr|ξi − ξ| ≥ x− z.

Since Pr|ξi − ξ| ≥ x − z → 0, we obtain Φ(z) ≤ lim infi→∞Φi(x) for anyz < x. Letting z → x, we get

Φ(x) ≤ lim infi→∞

Φi(x). (1.62)

It follows from (1.61) and (1.62) that Φi(x) → Φ(x). The theorem is proved.

Example 1.32: Convergence in distribution does not imply convergencein probability. For example, take (Ω,A,Pr) to be ω1, ω2 with Prω1 =Prω2 = 0.5, and

ξ(ω) =−1, if ω = ω1

1, if ω = ω2.

We also define ξi = −ξ for all i. Then ξi and ξ are identically distributed.Thus ξi converges in distribution to ξ. But, for any small number ε > 0,we have Pr|ξi − ξ| > ε = PrΩ = 1. That is, the sequence ξi does notconverge in probability to ξ.


1.14 Conditional Probability

We consider the probability of an event A after it has been learned thatsome other event B has occurred. This new probability of A is called theconditional probability of A given B.

Definition 1.28 Let (Ω,A,Pr) be a probability space, and A,B ∈ A. Thenthe conditional probability of A given B is defined by

PrA|B =PrA ∩B

PrB(1.63)

provided that PrB > 0.

Theorem 1.44 Let (Ω,A,Pr) be a probability space, and B an event withPrB > 0. Then Pr·|B defined by (1.63) is a probability measure, and(Ω,A,Pr·|B) is a probability space.

Proof: It is sufficient to prove that Pr·|B satisfies the normality, nonneg-ativity and countable additivity axioms. At first, we have

PrΩ|B =PrΩ ∩B

PrB=

PrBPrB

= 1.

Secondly, for any A ∈ A, the set function PrA|B is nonnegative. Finally,for any countable sequence Ai of mutually disjoint events, we have

Pr

∞⋃i=1

Ai|B

=

Pr( ∞⋃

i=1

Ai

)∩B

PrB

=

∞∑i=1

PrAi ∩B

PrB=

∞∑i=1

PrAi|B.

Thus Pr·|B is a probability measure. Furthermore, (Ω,A,Pr·|B) is aprobability space.

Theorem 1.45 (Bayes Formula) Let the events A1, A2, · · · , An form a par-tition of the space Ω such that PrAi > 0 for i = 1, 2, · · · , n, and let B bean event with PrB > 0. Then we have

PrAk|B =PrAkPrB|Akn∑

i=1

PrAiPrB|Ai(1.64)

for k = 1, 2, · · · , n.

Proof: Since A1, A2, · · · , An form a partition of the space Ω, we have

PrB =n∑

i=1

PrAi ∩B =n∑

i=1

PrAiPrB|Ai


which is also called the formula for total probability. Thus, for any k, we have

PrAk|B =PrAk ∩B

PrB=

PrAkPrB|Akn∑

i=1

PrAiPrB|Ai.


Remark 1.2: Especially, let A and B be two events with PrA > 0 andPrB > 0. Then A and Ac form a partition of the space Ω, and the Bayesformula is

PrA|B =PrAPrB|A

PrB. (1.65)

Remark 1.3: In statistical applications, the events A1, A2, · · · , An are oftencalled hypotheses. Furthermore, for each i, the PrAi is called the priorprobability of Ai, and PrAi|B is called the posterior probability of Ai afterthe occurrence of event B.

Example 1.33: Let ξ be an exponentially distributed random variable withexpected value β. Then for any real numbers a > 0 and x > 0, the conditionalprobability of ξ ≥ a+ x given ξ ≥ a is

Prξ ≥ a+ x|ξ ≥ a = exp(−x/β) = Prξ ≥ x

which means that the conditional probability is identical to the original prob-ability. This is the so-called memoryless property of exponential distribution.In other words, it is as good as new if it is functioning on inspection.

Definition 1.29 Let ξ be a random variable on (Ω,A,Pr). A conditionalrandom variable of ξ given B is a measurable function ξ|B from the condi-tional probability space (Ω,A,Pr·|B) to the set of real numbers such that

ξ|B(ω) ≡ ξ(ω), ∀ω ∈ Ω. (1.66)

Definition 1.30 The conditional probability distribution Φ: < → [0, 1] of arandom variable ξ given B is defined by

Φ(x|B) = Pr ξ ≤ x|B (1.67)

provided that PrB > 0.

Example 1.34: Let ξ and η be random variables. Then the conditionalprobability distribution of ξ given η = y is

Φ(x|η = y) = Pr ξ ≤ x|η = y =Prξ ≤ x, η = y

Prη = y

provided that Prη = y > 0.


Definition 1.31 The conditional probability density function φ of a randomvariable ξ given B is a nonnegative function such that

Φ(x|B) =∫ x

−∞φ(y|B)dy, ∀x ∈ < (1.68)

where Φ(x|B) is the conditional probability distribution of ξ given B.

Example 1.35: Let (ξ, η) be a random vector with joint probability densityfunction ψ. Then the marginal probability density functions of ξ and η are

f(x) =∫ +∞

−∞ψ(x, y)dy, g(y) =

∫ +∞

−∞ψ(x, y)dx,

respectively. Furthermore, we have

Prξ ≤ x, η ≤ y =∫ x

−∞

∫ y

−∞ψ(r, t)drdt =

∫ y

−∞

[∫ x

−∞

ψ(r, t)g(t)

dr]g(t)dt

which implies that the conditional probability distribution of ξ given η = yis

Φ(x|η = y) =∫ x

−∞

ψ(r, y)g(y)

dr, a.s. (1.69)

and the conditional probability density function of ξ given η = y is

φ(x|η = y) =ψ(x, y)g(y)

=ψ(x, y)∫ +∞

−∞ψ(x, y)dx

, a.s. (1.70)

Note that (1.69) and (1.70) are defined only for g(y) 6= 0. In fact, the sety|g(y) = 0 has probability 0. Especially, if ξ and η are independent randomvariables, then ψ(x, y) = f(x)g(y) and φ(x|η = y) = f(x).

Definition 1.32 Let ξ be a random variable. Then the conditional expectedvalue of ξ given B is defined by

E[ξ|B] =∫ +∞

0

Prξ ≥ r|Bdr −∫ 0

−∞Prξ ≤ r|Bdr (1.71)


Following conditional probability and conditional expected value, we alsohave conditional variance, conditional moments, conditional critical values,conditional entropy as well as conditional convergence.


Hazard Rate

Definition 1.33 Let ξ be a nonnegative random variable representing life-time. Then the hazard rate (or failure rate) is

h(x) = lim∆↓0

Prξ ≤ x+ ∆∣∣ ξ > x

∆. (1.72)

The hazard rate tells us the probability of a failure just after time x whenit is functioning at time x. If ξ has probability distribution Φ and probabilitydensity function φ, then the hazard rate

h(x) =φ(x)

1− Φ(x).

Example 1.36: Let ξ be an exponentially distributed random variable withexpected value β. Then its hazard rate h(x) ≡ 1/β.

Chapter 2

Credibility Theory

The concept of fuzzy set was initiated by Zadeh [259] via membership functionin 1965. In order to measure a fuzzy event, Zadeh [262] proposed the conceptof possibility measure. Although possibility measure has been widely used,it has no self-duality property. However, a self-dual measure is absolutelyneeded in both theory and practice. In order to define a self-dual measure,Liu and Liu [132] presented the concept of credibility measure. In addition,a sufficient and necessary condition for credibility measure was given by Liand Liu [102]. Credibility theory, founded by Liu [135] in 2004 and refinedby Liu [138] in 2007, is a branch of mathematics for studying the behavior offuzzy phenomena.

The emphasis in this chapter is mainly on credibility measure, credibilityspace, fuzzy variable, membership function, credibility distribution, indepen-dence, identical distribution, expected value, variance, moments, critical val-ues, entropy, distance, convergence almost surely, convergence in credibility,convergence in mean, convergence in distribution, and conditional credibility.

2.1 Credibility Space

Let Θ be a nonempty set, and P the power set of Θ (i.e., the larggest σ-algebra over Θ). Each element in P is called an event. In order to present anaxiomatic definition of credibility, it is necessary to assign to each event A anumber CrA which indicates the credibility that A will occur. In order toensure that the number CrA has certain mathematical properties which weintuitively expect a credibility to have, we accept the following four axioms:

Axiom 1. (Normality) CrΘ = 1.

Axiom 2. (Monotonicity) CrA ≤ CrB whenever A ⊂ B.

Axiom 3. (Self-Duality) CrA+ CrAc = 1 for any event A.

46 Chapter 2 - Credibility Theory

Axiom 4. (Maximality) Cr ∪iAi = supi CrAi for any events Ai withsupi CrAi < 0.5.

Definition 2.1 (Liu and Liu [132]) The set function Cr is called a cred-ibility measure if it satisfies the normality, monotonicity, self-duality, andmaximality axioms.

Example 2.1: Let Θ = θ1, θ2. For this case, there are only four events:∅, θ1, θ2,Θ. Define Cr∅ = 0, Crθ1 = 0.7, Crθ2 = 0.3, and CrΘ =1. Then the set function Cr is a credibility measure because it satisfies thefour axioms.

Example 2.2: Let Θ be a nonempty set. Define Cr∅ = 0, CrΘ = 1 andCrA = 1/2 for any subset A (excluding ∅ and Θ). Then the set functionCr is a credibility measure.

Example 2.3: Let µ be a nonnegative function on Θ (for example, the setof real numbers) such that

supx∈Θ

µ(x) = 1. (2.1)

Then the set function

CrA =12

(supx∈A

µ(x) + 1− supx∈Ac

µ(x))

(2.2)

is a credibility measure on Θ.

Theorem 2.1 Let Θ be a nonempty set, P the power set of Θ, and Cr thecredibility measure. Then Cr∅ = 0 and 0 ≤ CrA ≤ 1 for any A ∈ P.

Proof: It follows from Axioms 1 and 3 that Cr∅ = 1−CrΘ = 1−1 = 0.Since ∅ ⊂ A ⊂ Θ, we have 0 ≤ CrA ≤ 1 by using Axiom 2.

Theorem 2.2 Let Θ be a nonempty set, P the power set of Θ, and Cr thecredibility measure. Then for any A,B ∈ P, we have

CrA ∪B = CrA ∨ CrB if CrA ∪B ≤ 0.5, (2.3)

CrA ∩B = CrA ∧ CrB if CrA ∩B ≥ 0.5. (2.4)

The above equations hold for not only finite number of events but also infinitenumber of events.

Proof: If CrA ∪ B < 0.5, then CrA ∨ CrB < 0.5 by using Axiom 2.Thus the equation (2.3) follows immediately from Axiom 4. If CrA ∪B =0.5 and (2.3) does not hold, then we have CrA ∨ CrB < 0.5. It followsfrom Axiom 4 that

CrA ∪B = CrA ∨ CrB < 0.5.

Section 2.1 - Credibility Space 47

A contradiction proves (2.3). Next we prove (2.4). Since CrA ∩ B ≥ 0.5,we have CrAc ∪Bc ≤ 0.5 by the self-duality. Thus

CrA ∩B = 1− CrAc ∪Bc = 1− CrAc ∨ CrBc

= (1− CrAc) ∧ (1− CrBc) = CrA ∧ CrB.


Theorem 2.3 Let Θ be a nonempty set, P the power set of Θ, and Cr thecredibility measure. Then for any A,B ∈ P, we have

CrA ∪B = CrA ∨ CrB if CrA+ CrB < 1, (2.5)

CrA ∩B = CrA ∧ CrB if CrA+ CrB > 1. (2.6)

Proof: Suppose CrA + CrB < 1. Then there exists at least one termless than 0.5, say CrB < 0.5. If CrA < 0.5 also holds, then the equation(2.3) follows immediately from Axiom 4. If CrA ≥ 0.5, then by usingTheorem 2.2, we obtain

CrA = CrA∪(B∩Bc) = Cr(A∪B)∩(A∪Bc) = CrA∪B∧CrA∪Bc.

On the other hand, we have

CrA < 1− CrB = CrBc ≤ CrA ∪Bc.

Hence we must have CrA ∪ B = CrA = CrA ∨ CrB. The equation(2.5) is proved. Next we suppose CrA + CrB > 1. Then CrAc +CrBc < 1. It follows from (2.5) that

CrA ∩B = 1− CrAc ∪Bc = 1− CrAc ∨ CrBc

= (1− CrAc) ∧ (1− CrBc) = CrA ∧ CrB.


Credibility Subadditivity Theorem

Theorem 2.4 (Liu [135], Credibility Subadditivity Theorem) The credibilitymeasure is subadditive. That is,

CrA ∪B ≤ CrA+ CrB (2.7)

for any events A and B. In fact, credibility measure is not only finitelysubadditive but also countably subadditive.


Proof: The argument breaks down into three cases.Case 1: CrA < 0.5 and CrB < 0.5. It follows from Axiom 4 that

CrA ∪B = CrA ∨ CrB ≤ CrA+ CrB.

Case 2: CrA ≥ 0.5. For this case, by using Axioms 2 and 3, we haveCrAc ≤ 0.5 and CrA ∪B ≥ CrA ≥ 0.5. Then

CrAc = CrAc ∩B ∨ CrAc ∩Bc

≤ CrAc ∩B+ CrAc ∩Bc

≤ CrB+ CrAc ∩Bc.

Applying this inequality, we obtain

CrA+ CrB = 1− CrAc+ CrB

≥ 1− CrB − CrAc ∩Bc+ CrB

= 1− CrAc ∩Bc

= CrA ∪B.

Case 3: CrB ≥ 0.5. This case may be proved by a similar process ofCase 2. The theorem is proved.

Remark 2.1: For any eventsA andB, it follows from the credibility subaddi-tivity theorem that the credibility measure is null-additive, i.e., CrA∪B =CrA+ CrB if either CrA = 0 or CrB = 0.

Theorem 2.5 Let Bi be a decreasing sequence of events with CrBi → 0as i→∞. Then for any event A, we have

limi→∞

CrA ∪Bi = limi→∞

CrA\Bi = CrA. (2.8)

Proof: It follows from the monotonicity axiom and credibility subadditivitytheorem that

CrA ≤ CrA ∪Bi ≤ CrA+ CrBi

for each i. Thus we get CrA ∪ Bi → CrA by using CrBi → 0. Since(A\Bi) ⊂ A ⊂ ((A\Bi) ∪Bi), we have

CrA\Bi ≤ CrA ≤ CrA\Bi+ CrBi.

Hence CrA\Bi → CrA by using CrBi → 0.

Theorem 2.6 A credibility measure on Θ is additive if and only if there areat most two elements in Θ taking nonzero credibility values.


Proof: Suppose that the credibility measure is additive. If there are morethan two elements taking nonzero credibility values, then we may choose threeelements θ1, θ2, θ3 such that Crθ1 ≥ Crθ2 ≥ Crθ3 > 0. If Crθ1 ≥ 0.5,it follows from Axioms 2 and 3 that

Crθ2, θ3 ≤ CrΘ \ θ1 = 1− Crθ1 ≤ 0.5.

By using Axiom 4, we obtain

Crθ2, θ3 = Crθ2 ∨ Crθ3 < Crθ2+ Crθ3.

This is in contradiction with the additivity assumption. If Crθ1 < 0.5,then Crθ3 ≤ Crθ2 < 0.5. It follows from Axiom 4 that

Crθ2, θ3 ∧ 0.5 = Crθ2 ∨ Crθ3 < 0.5

which implies that

Crθ2, θ3 = Crθ2 ∨ Crθ3 < Crθ2+ Crθ3.

This is also in contradiction with the additivity assumption. Hence there areat most two elements taking nonzero credibility values.

Conversely, suppose that there are at most two elements, say θ1 and θ2,taking nonzero credibility values. Let A and B be two disjoint events. Theargument breaks down into two cases.

Case 1: CrA = 0 or CrB = 0. Then we have CrA ∪B = CrA+CrB by using the credibility subadditivity theorem.

Case 2: CrA > 0 or CrB > 0. For this case, without loss of generality,we suppose that θ1 ∈ A and θ2 ∈ B. Note that Cr(A ∪B)c = 0. It followsfrom Axiom 3 and the credibility subadditivity theorem that

CrA ∪B = CrA ∪B ∪ (A ∪B)c = CrΘ = 1,

CrA+ CrB = CrA ∪ (A ∪B)c+ CrB = 1.

Hence CrA ∪B = CrA+ CrB. The additivity is proved.

Remark 2.2: Theorem 2.6 states that a credibility measure is identical withprobability measure if there are effectively two elements in the universal set.

Credibility Semicontinuity Law

Generally speaking, the credibility measure is neither lower semicontinuousnor upper semicontinuous. However, we have the following credibility semi-continuity law.

Theorem 2.7 (Liu [135], Credibility Semicontinuity Law) For any eventsA1, A2, · · · , we have

limi→∞

CrAi = Cr

limi→∞

Ai

(2.9)


if one of the following conditions is satisfied:(a) Cr A ≤ 0.5 and Ai ↑ A; (b) lim

i→∞CrAi < 0.5 and Ai ↑ A;

(c) Cr A ≥ 0.5 and Ai ↓ A; (d) limi→∞

CrAi > 0.5 and Ai ↓ A.

Proof: (a) Since CrA ≤ 0.5, we have CrAi ≤ 0.5 for each i. It followsfrom Axiom 4 that

CrA = Cr ∪iAi = supi

CrAi = limi→∞

CrAi.

(b) Since limi→∞CrAi < 0.5, we have supi CrAi < 0.5. It followsfrom Axiom 4 that

CrA = Cr ∪iAi = supi

CrAi = limi→∞

CrAi.

(c) Since CrA ≥ 0.5 and Ai ↓ A, it follows from the self-duality ofcredibility measure that CrAc ≤ 0.5 and Ac

i ↑ Ac. Thus CrAi = 1 −CrAc

i → 1− CrAc = CrA as i→∞.(d) Since limi→∞CrAi > 0.5 and Ai ↓ A, it follows from the self-duality

of credibility measure that

limi→∞

CrAci = lim

i→∞(1− CrAi) < 0.5

and Aci ↑ Ac. Thus CrAi = 1−CrAc

i → 1−CrAc = CrA as i→∞.The theorem is proved.

Credibility Asymptotic Theorem

Theorem 2.8 (Credibility Asymptotic Theorem) For any events A1, A2, · · · ,we have

limi→∞

CrAi ≥ 0.5, if Ai ↑ Θ, (2.10)

limi→∞

CrAi ≤ 0.5, if Ai ↓ ∅. (2.11)

Proof: Assume Ai ↑ Θ. If limi→∞CrAi < 0.5, it follows from the credi-bility semicontinuity law that

CrΘ = limi→∞

CrAi < 0.5

which is in contradiction with CrΘ = 1. The first inequality is proved.The second one may be verified similarly.


Credibility Extension Theorem

Suppose that the credibility of each singleton is given. Is the credibilitymeasure fully and uniquely determined? This subsection will answer thequestion.

Theorem 2.9 Suppose that Θ is a nonempty set. If Cr is a credibility mea-sure, then we have

supθ∈Θ

Crθ ≥ 0.5,

Crθ∗+ supθ 6=θ∗

Crθ = 1 if Crθ∗ ≥ 0.5.(2.12)

We will call (2.12) the credibility extension condition.

Proof: If supCrθ < 0.5, then by using Axiom 4, we have

1 = CrΘ = supθ∈Θ

Crθ < 0.5.

This contradiction proves supCrθ ≥ 0.5. We suppose that θ∗ ∈ Θ is a pointwith Crθ∗ ≥ 0.5. It follows from Axioms 3 and 4 that CrΘ \ θ∗ ≤ 0.5,and

CrΘ \ θ∗ = supθ 6=θ∗

Crθ.

Hence the second formula of (2.12) is true by the self-duality of credibilitymeasure.

Theorem 2.10 (Li and Liu [102], Credibility Extension Theorem) Supposethat Θ is a nonempty set, and Crθ is a nonnegative function on Θ satisfyingthe credibility extension condition (2.12). Then Crθ has a unique extensionto a credibility measure as follows,

CrA =

supθ∈A

Crθ, if supθ∈A

Crθ < 0.5

1− supθ∈Ac

Crθ, if supθ∈A

Crθ ≥ 0.5.(2.13)

Proof: We first prove that the set function CrA defined by (2.13) is acredibility measure.

Step 1: By the credibility extension condition supθ∈Θ

Crθ ≥ 0.5, we have

CrΘ = 1− supθ∈∅

Crθ = 1− 0 = 1.

Step 2: If A ⊂ B, then Bc ⊂ Ac. The proof breaks down into two cases.Case 1: sup

θ∈ACrθ < 0.5. For this case, we have

CrA = supθ∈A

Crθ ≤ supθ∈B

Crθ ≤ CrB.


Case 2: supθ∈A

Crθ ≥ 0.5. For this case, we have supθ∈B

Crθ ≥ 0.5, and

CrA = 1− supθ∈Ac

Crθ ≤ 1− supθ∈Bc

Crθ = CrB.

Step 3: In order to prove CrA + CrAc = 1, the argument breaksdown into two cases.

Case 1: supθ∈A

Crθ < 0.5. For this case, we have supθ∈Ac

Crθ ≥ 0.5. Thus,

CrA+ CrAc = supθ∈A

Crθ+ 1− supθ∈A

Crθ = 1.

Case 2: supθ∈A

Crθ ≥ 0.5. For this case, we have supθ∈Ac

Crθ ≤ 0.5, and

CrA+ CrAc = 1− supθ∈Ac

Crθ+ supθ∈Ac

Crθ = 1.

Step 4: For any collection Ai with supi CrAi < 0.5, we have

Cr∪iAi = supθ∈∪iAi

Crθ = supi

supθ∈Ai

Crθ = supi

CrAi.

Thus Cr is a credibility measure because it satisfies the four axioms.Finally, let us prove the uniqueness. Assume that Cr1 and Cr2 are two

credibility measures such that Cr1θ = Cr2θ for each θ ∈ Θ. Let us provethat Cr1A = Cr2A for any event A. The argument breaks down intothree cases.

Case 1: Cr1A < 0.5. For this case, it follows from Axiom 4 that

Cr1A = supθ∈A

Cr1θ = supθ∈A

Cr2θ = Cr2A.

Case 2: Cr1A > 0.5. For this case, we have Cr1Ac < 0.5. It followsfrom the first case that Cr1Ac = Cr2Ac which implies Cr1A = Cr2A.

Case 3: Cr1A = 0.5. For this case, we have Cr1Ac = 0.5, and

Cr2A ≥ supθ∈A

Cr2θ = supθ∈A

Cr1θ = Cr1A = 0.5,

Cr2Ac ≥ supθ∈Ac

Cr2θ = supθ∈Ac

Cr1θ = Cr1Ac = 0.5.

Hence Cr2A = 0.5 = Cr1A. The uniqueness is proved.

Credibility Space

Definition 2.2 Let Θ be a nonempty set, P the power set of Θ, and Cr acredibility measure. Then the triplet (Θ,P,Cr) is called a credibility space.


Example 2.4: The triplet (Θ,P,Cr) is a credibility space if

Θ = θ1, θ2, · · · , Crθi ≡ 1/2 for i = 1, 2, · · · (2.14)

Note that the credibility measure is produced by the credibility extensiontheorem as follows,

CrA =

0, if A = ∅1, if A = Θ

1/2, otherwise.


Θ = θ1, θ2, · · · , Crθi = i/(2i+ 1) for i = 1, 2, · · · (2.15)

By using the credibility extension theorem, we obtain the following credibilitymeasure,

CrA =

supθi∈A

i

2i+ 1, if A is finite

1− supθi∈Ac

i

2i+ 1, if A is infinite.


Θ = θ1, θ2, · · · , Crθ1 = 1/2, Crθi = 1/i for i = 2, 3, · · · (2.16)

For this case, the credibility measure is

CrA =

supθi∈A

1/i, if A contains neither θ1 nor θ2

1/2, if A contains only one of θ1 and θ2

1− supθi∈Ac

1/i, if A contains both θ1 and θ2.


Θ = [0, 1], Crθ = θ/2 for θ ∈ Θ. (2.17)

For this case, the credibility measure is

CrA =

12

supθ∈A

θ, if supθ∈A

θ < 1

1− 12

supθ∈Ac

θ, if supθ∈A

θ = 1.


Product Credibility Measure

Product credibility measure may be defined in multiple ways. This bookaccepts the following axiom.

Axiom 5. (Product Credibility Axiom) Let Θk be nonempty sets on whichCrk are credibility measures, k = 1, 2, · · · , n, respectively, and Θ = Θ1×Θ2×· · · ×Θn. Then

Cr(θ1, θ2, · · · , θn) = Cr1θ1 ∧ Cr2θ2 ∧ · · · ∧ Crnθn (2.18)

for each (θ1, θ2, · · · , θn) ∈ Θ.

Theorem 2.11 (Product Credibility Theorem) Let Θk be nonempty sets onwhich Crk are the credibility measures, k = 1, 2, · · · , n, respectively, and Θ =Θ1×Θ2×· · ·×Θn. Then Cr = Cr1 ∧Cr2 ∧ · · · ∧Crn defined by Axiom 5 hasa unique extension to a credibility measure on Θ as follows,

CrA =

sup(θ1,θ2··· ,θn)∈A

min1≤k≤n

Crkθk,

if sup(θ1,θ2,··· ,θn)∈A

min1≤k≤n

Crkθk < 0.5

1− sup(θ1,θ2,··· ,θn)∈Ac

min1≤k≤n

Crkθk,

if sup(θ1,θ2,··· ,θn)∈A

min1≤k≤n

Crkθk ≥ 0.5.

(2.19)

Proof: For each θ = (θ1, θ2, · · · , θn) ∈ Θ, we have Crθ = Cr1θ1 ∧Cr2θ2 ∧ · · · ∧ Crnθn. Let us prove that Crθ satisfies the credibilityextension condition. Since sup

θk∈Θk

Crθk ≥ 0.5 for each k, we have

supθ∈Θ

Crθ = sup(θ1,θ2,··· ,θn)∈Θ

min1≤k≤n

Crkθk ≥ 0.5.

Now we suppose that θ∗ = (θ∗1 , θ∗2 , · · · , θ∗n) is a point with Crθ∗ ≥ 0.5.

Without loss of generality, let i be the index such that

Crθ∗ = min1≤k≤n

Crkθ∗k = Criθ∗i . (2.20)

We also immediately have

Crkθ∗k ≥ 0.5, k = 1, 2, · · · , n; (2.21)

Crkθ∗k+ supθk 6=θ∗k

Crkθk = 1, k = 1, 2, · · · , n; (2.22)

supθi 6=θ∗i

Criθi ≥ supθk 6=θ∗k

Crkθk, k = 1, 2, · · · , n; (2.23)


supθk 6=θ∗k

Crkθk ≤ 0.5, k = 1, · · · , n. (2.24)

It follows from (2.21) and (2.24) that

supθ 6=θ∗

Crθ = sup(θ1,θ2,··· ,θn) 6=(θ∗1 ,θ∗2 ,··· ,θ∗n)

min1≤k≤n

Crkθk

≥ supθi 6=θ∗i

min1≤k≤i−1

Crkθ∗k ∧ Criθi ∧ mini+1≤k≤n

Crkθ∗k

= supθi 6=θ∗i

Criθi.

We next suppose that

supθ 6=θ∗

Crθ > supθi 6=θ∗i

Criθi.

Then there is a point (θ′1, θ′2, · · · , θ′n) 6= (θ∗1 , θ

∗2 , · · · , θ∗n) such that

min1≤k≤n

Crkθ′k > supθi 6=θ∗i

Criθi.

Let j be one of the index such that θ′j 6= θ∗j . Then

Crjθ′j > supθi 6=θ∗i

Criθi.

That is,sup

θj 6=θ∗j

Crjθj > supθi 6=θ∗i

Criθi

which is in contradiction with (2.23). Thus

supθ 6=θ∗

Crθ = supθi 6=θ∗i

Criθi. (2.25)

It follows from (2.20), (2.22) and (2.25) that

Crθ∗+ supθ 6=θ∗

Crθ = Criθ∗i + supθi 6=θ∗i

Criθi = 1.

Thus Cr satisfies the credibility extension condition. It follows from the cred-ibility extension theorem that CrA is just the unique extension of Crθ.The theorem is proved.

Definition 2.3 Let (Θk,Pk,Crk), k = 1, 2, · · · , n be credibility spaces, Θ =Θ1×Θ2× · · · ×Θn and Cr = Cr1 ∧Cr2 ∧ · · · ∧Crn. Then (Θ,P,Cr) is calledthe product credibility space of (Θk,Pk,Crk), k = 1, 2, · · · , n.

Theorem 2.12 Let (Θ,P,Cr) be the product credibility space of (Θk,Pk,Crk),k = 1, 2, · · · , n. Then for any Ak ∈ Pk, k = 1, 2, · · · , n, we have

CrA1 ×A2 × · · · ×Ak = Cr1A1 ∧ Cr2A2 ∧ · · · ∧ CrnAn.


Proof: We only prove the case of n = 2. If Cr1A1 < 0.5 or Cr2A2 < 0.5,then we have

supθ1∈A1

Cr1θ1 < 0.5 or supθ2∈A2

Cr2θ2 < 0.5.

It follows from

sup(θ1,θ2)∈A1×A2

Cr1θ1 ∧ Cr2θ2 = supθ1∈A1

Cr1θ1 ∧ supθ2∈A2

Cr2θ2 < 0.5

that

CrA1 ×A2 = supθ1∈A1


Cr2θ2 = Cr1A1 ∧ Cr2A2.

If Cr1A1 ≥ 0.5 and Cr2A2 ≥ 0.5, then we have

supθ1∈A1

Cr1θ1 ≥ 0.5 and supθ2∈A2

Cr2θ2 ≥ 0.5.

It follows from

sup(θ1,θ2)∈A1×A2

Cr1θ1 ∧ Cr2θ2 = supθ1∈A1


Cr2θ2 ≥ 0.5

thatCrA1 ×A2 = 1− sup

(θ1,θ2)/∈A1×A2

Cr1θ1 ∧ Cr2θ2

=

(1− sup

θ1∈Ac1

Cr1θ1

)∧

(1− sup

θ2∈Ac2

Cr2θ2

)= Cr1A1 ∧ Cr2A2.


Theorem 2.13 (Infinite Product Credibility Theorem) Suppose that Θk arenonempty sets, Crk the credibility measures on Pk, k = 1, 2, · · · , respectively.Let Θ = Θ1 ×Θ2 × · · · Then

CrA =

sup(θ1,θ2,··· )∈A

inf1≤k<∞

Crkθk,

if sup(θ1,θ2,··· )∈A

inf1≤k<∞

Crkθk < 0.5

1− sup(θ1,θ2,··· )∈Ac

inf1≤k<∞

Crkθk,

if sup(θ1,θ2,··· )∈A

inf1≤k<∞

Crkθk ≥ 0.5

is a credibility measure on P.

Proof: Like Theorem 2.11 except Cr(θ1, θ2, · · · ) = inf1≤k<∞Crkθk.Definition 2.4 Let (Θk,Pk,Crk), k = 1, 2, · · · be credibility spaces. DefineΘ = Θ1 × Θ2 × · · · and Cr = Cr1 ∧ Cr2 ∧ · · · Then (Θ,P,Cr) is called theinfinite product credibility space of (Θk,Pk,Crk), k = 1, 2, · · ·

Section 2.2 - Fuzzy Variables 57

2.2 Fuzzy Variables

Definition 2.5 A fuzzy variable is a (measurable) function from a credibilityspace (Θ,P,Cr) to the set of real numbers.

Example 2.8: Take (Θ,P,Cr) to be θ1, θ2 with Crθ1 = Crθ2 = 0.5.Then the function

ξ(θ) =

0, if θ = θ1

1, if θ = θ2

is a fuzzy variable.

Example 2.9: Take (Θ,P,Cr) to be the interval [0, 1] with Crθ = θ/2 foreach θ ∈ [0, 1]. Then the identity function ξ(θ) = θ is a fuzzy variable.

Example 2.10: A crisp number c may be regarded as a special fuzzy vari-able. In fact, it is the constant function ξ(θ) ≡ c on the credibility space(Θ,P,Cr).

Remark 2.3: Since a fuzzy variable ξ is a function on a credibility space,for any set B of real numbers, the set

ξ ∈ B =θ ∈ Θ

∣∣ ξ(θ) ∈ B (2.26)

is always an element in P. In other words, the fuzzy variable ξ is always ameasurable function and ξ ∈ B is always an event.

Definition 2.6 A fuzzy variable ξ is said to be(a) nonnegative if Crξ < 0 = 0;(b) positive if Crξ ≤ 0 = 0;(c) continuous if Crξ = x is a continuous function of x;(d) simple if there exists a finite sequence x1, x2, · · · , xm such that

Cr ξ 6= x1, ξ 6= x2, · · · , ξ 6= xm = 0; (2.27)


Cr ξ 6= x1, ξ 6= x2, · · · = 0. (2.28)

Definition 2.7 Let ξ1 and ξ2 be fuzzy variables defined on the credibilityspace (Θ,P,Cr). We say ξ1 = ξ2 if ξ1(θ) = ξ2(θ) for almost all θ ∈ Θ.

Fuzzy Vector

Definition 2.8 An n-dimensional fuzzy vector is defined as a function froma credibility space (Θ,P,Cr) to the set of n-dimensional real vectors.

Theorem 2.14 The vector (ξ1, ξ2, · · · , ξn) is a fuzzy vector if and only ifξ1, ξ2, · · · , ξn are fuzzy variables.


Proof: Write ξ = (ξ1, ξ2, · · · , ξn). Suppose that ξ is a fuzzy vector. Thenξ1, ξ2, · · · , ξn are functions from Θ to <. Thus ξ1, ξ2, · · · , ξn are fuzzy vari-ables. Conversely, suppose that ξi are fuzzy variables defined on the cred-ibility spaces (Θi,Pi,Cri), i = 1, 2, · · · , n, respectively. It is clear that(ξ1, ξ2, · · · , ξn) is a function from the product credibility space (Θ,P,Cr)to <n, i.e.,

ξ(θ1, θ2, · · · , θn) = (ξ1(θ1), ξ2(θ2), · · · , ξn(θn))

for all (θ1, θ2, · · · , θn) ∈ Θ. Hence ξ = (ξ1, ξ2, · · · , ξn) is a fuzzy vector.

Fuzzy Arithmetic

In this subsection, we will suppose that all fuzzy variables are defined on acommon credibility space. Otherwise, we may embed them into the productcredibility space.

Definition 2.9 Let f : <n → < be a function, and ξ1, ξ2, · · · , ξn fuzzy vari-ables on the credibility space (Θ,P,Cr). Then ξ = f(ξ1, ξ2, · · · , ξn) is a fuzzyvariable defined as

ξ(θ) = f(ξ1(θ), ξ2(θ), · · · , ξn(θ)) (2.29)

for any θ ∈ Θ.

Example 2.11: Let ξ1 and ξ2 be fuzzy variables on the credibility space(Θ,P,Cr). Then their sum is

(ξ1 + ξ2)(θ) = ξ1(θ) + ξ2(θ), ∀θ ∈ Θ

and their product is

(ξ1 × ξ2)(θ) = ξ1(θ)× ξ2(θ), ∀θ ∈ Θ.

The reader may wonder whether ξ(θ1, θ2, · · · , θn) defined by (2.29) is afuzzy variable. The following theorem answers this question.

Theorem 2.15 Let ξ be an n-dimensional fuzzy vector, and f : <n → < afunction. Then f(ξ) is a fuzzy variable.

Proof: Since f(ξ) is a function from a credibility space to the set of realnumbers, it is a fuzzy variable.

2.3 Membership Function

Definition 2.10 Let ξ be a fuzzy variable defined on the credibility space(Θ,P,Cr). Then its membership function is derived from the credibility mea-sure by

µ(x) = (2Crξ = x) ∧ 1, x ∈ <. (2.30)

Section 2.3 - Membership Function 59

Membership function represents the degree that the fuzzy variable ξ takessome prescribed value. How do we determine membership functions? Thereare several methods reported in the past literature. Anyway, the membershipdegree µ(x) = 0 if x is an impossible point, and µ(x) = 1 if x is the mostpossible point that ξ takes.

Example 2.12: It is clear that a fuzzy variable has a unique membershipfunction. However, a membership function may produce multiple fuzzy vari-ables. For example, let Θ = θ1, θ2 and Crθ1 = Crθ2 = 0.5. Then(Θ,P,Cr) is a credibility space. We define

ξ1(θ) =

0, if θ = θ11, if θ = θ2,

ξ2(θ) =

1, if θ = θ10, if θ = θ2.

It is clear that both of them are fuzzy variables and have the same member-ship function, µ(x) ≡ 1 on x = 0 or 1.

Theorem 2.16 (Credibility Inversion Theorem) Let ξ be a fuzzy variablewith membership function µ. Then for any set B of real numbers, we have

Crξ ∈ B =12

(supx∈B

µ(x) + 1− supx∈Bc

µ(x)). (2.31)

Proof: If Crξ ∈ B ≤ 0.5, then by Axiom 2, we have Crξ = x ≤ 0.5 foreach x ∈ B. It follows from Axiom 4 that

Crξ ∈ B =12

(supx∈B

(2Crξ = x ∧ 1))

=12

supx∈B

µ(x). (2.32)

The self-duality of credibility measure implies that Crξ ∈ Bc ≥ 0.5 andsupx∈Bc Crξ = x ≥ 0.5, i.e.,

supx∈Bc

µ(x) = supx∈Bc

(2Crξ = x ∧ 1) = 1. (2.33)

It follows from (2.32) and (2.33) that (2.31) holds.If Crξ ∈ B ≥ 0.5, then Crξ ∈ Bc ≤ 0.5. It follows from the first case

that

Crξ ∈ B = 1− Crξ ∈ Bc = 1− 12

(sup

x∈Bc

µ(x) + 1− supx∈B

µ(x))

=12

(supx∈B


µ(x)).


Example 2.13: Let ξ be a fuzzy variable with membership function µ. Thenthe following equations follow immediately from Theorem 2.16:

Crξ = x =12

(µ(x) + 1− sup

y 6=xµ(y)

), ∀x ∈ <; (2.34)


Crξ ≤ x =12

(supy≤x

µ(y) + 1− supy>x

µ(y)), ∀x ∈ <; (2.35)

Crξ ≥ x =12

(supy≥x

µ(y) + 1− supy<x

µ(y)), ∀x ∈ <. (2.36)

Especially, if µ is a continuous function, then

Crξ = x =µ(x)

2, ∀x ∈ <. (2.37)

Theorem 2.17 (Sufficient and Necessary Condition for Membership Func-tion) A function µ : < → [0, 1] is a membership function if and only ifsupµ(x) = 1.

Proof: If µ is a membership function, then there exists a fuzzy variable ξwhose membership function is just µ, and

supx∈<

µ(x) = supx∈<

(2Crξ = x) ∧ 1.

If there is some point x ∈ < such that Crξ = x ≥ 0.5, then supµ(x) = 1.Otherwise, we have Crξ = x < 0.5 for each x ∈ <. It follows from Axiom 4that

supx∈<

µ(x) = supx∈<

(2Crξ = x) ∧ 1 = 2 supx∈<

Crξ = x = 2 (CrΘ ∧ 0.5) = 1.

Conversely, suppose that supµ(x) = 1. For each x ∈ <, we define

Crx =12

(µ(x) + 1− sup

y 6=xµ(y)

).

It is clear thatsupx∈<

Crx ≥ 12(1 + 1− 1) = 0.5.

For any x∗ ∈ < with Crx∗ ≥ 0.5, we have µ(x∗) = 1 and

Crx∗+ supy 6=x∗

Cry

=12

(µ(x∗) + 1− sup

y 6=x∗µ(y)

)+ sup

y 6=x∗

12

(µ(y) + 1− sup

z 6=yµ(z)

)

= 1− 12

supy 6=x∗

µ(y) +12

supy 6=x∗

µ(y) = 1.

Thus Crx satisfies the credibility extension condition, and has a uniqueextension to credibility measure on P(<) by using the credibility extension

Section 2.3 - Membership Function 61

theorem. Now we define a fuzzy variable ξ as an identity function from thecredibility space (<,P(<),Cr) to <. Then the membership function of thefuzzy variable ξ is

(2Crξ = x) ∧ 1 =

(µ(x) + 1− sup

y 6=xµ(y)

)∧ 1 = µ(x)

for each x. The theorem is proved.

Remark 2.4: Theorem 2.17 states that the identity function is a universalfunction for any fuzzy variable by defining an appropriate credibility space.

Theorem 2.18 A fuzzy variable ξ with membership function µ is(a) nonnegative if and only if µ(x) = 0 for all x < 0;(b) positive if and only if µ(x) = 0 for all x ≤ 0;(c) simple if and only if µ takes nonzero values at a finite number of points;(d) discrete if and only if µ takes nonzero values at a countable set of points;(e) continuous if and only if µ is a continuous function.

Proof: The theorem is obvious since the membership function µ(x) =(2Crξ = x) ∧ 1 for each x ∈ <.

Some Special Membership Functions

By an equipossible fuzzy variable we mean the fuzzy variable fully determinedby the pair (a, b) of crisp numbers with a < b, whose membership function isgiven by

µ1(x) =

1, if a ≤ x ≤ b

0, otherwise.

By a triangular fuzzy variable we mean the fuzzy variable fully determinedby the triplet (a, b, c) of crisp numbers with a < b < c, whose membershipfunction is given by

µ2(x) =

x− a

b− a, if a ≤ x ≤ b

x− c

b− c, if b ≤ x ≤ c

0, otherwise.

By a trapezoidal fuzzy variable we mean the fuzzy variable fully determinedby the quadruplet (a, b, c, d) of crisp numbers with a < b < c < d, whose


membership function is given by

µ3(x) =

x− a

b− a, if a ≤ x ≤ b

1, if b ≤ x ≤ c

x− d

c− d, if c ≤ x ≤ d

0, otherwise.

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µ1(x)

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µ3(x)

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Figure 2.1: Membership Functions µ1, µ2 and µ3

Joint Membership Function

Definition 2.11 If ξ = (ξ1, ξ2, · · · , ξn) is a fuzzy vector on the credibilityspace (Θ,P,Cr). Then its joint membership function is derived from thecredibility measure by

µ(x) = (2Crξ = x) ∧ 1, ∀x ∈ <n. (2.38)

Theorem 2.19 (Sufficient and Necessary Condition for Joint MembershipFunction) A function µ : <n → [0, 1] is a joint membership function if andonly if supµ(x) = 1.

Proof: Like Theorem 2.17.

2.4 Credibility Distribution

Definition 2.12 (Liu [130]) The credibility distribution Φ : < → [0, 1] of afuzzy variable ξ is defined by

Φ(x) = Crθ ∈ Θ

∣∣ ξ(θ) ≤ x. (2.39)

Section 2.4 - Credibility Distribution 63

That is, Φ(x) is the credibility that the fuzzy variable ξ takes a value less thanor equal to x. Generally speaking, the credibility distribution Φ is neitherleft-continuous nor right-continuous.

Example 2.14: The credibility distribution of an equipossible fuzzy variable(a, b) is

Φ1(x) =

0, if x < a

1/2, if a ≤ x < b

1, if x ≥ b.

Especially, if ξ is an equipossible fuzzy variable on <, then Φ1(x) ≡ 1/2.

Example 2.15: The credibility distribution of a triangular fuzzy variable(a, b, c) is

Φ2(x) =

0, if x ≤ ax− a

2(b− a), if a ≤ x ≤ b

x+ c− 2b2(c− b)

, if b ≤ x ≤ c

1, if x ≥ c.

Example 2.16: The credibility distribution of a trapezoidal fuzzy variable(a, b, c, d) is

Φ3(x) =

0, if x ≤ ax− a

2(b− a), if a ≤ x ≤ b

12, if b ≤ x ≤ c

x+ d− 2c2(d− c)

, if c ≤ x ≤ d

1, if x ≥ d.

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x

Φ1(x)

0

1

0.5

a b

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Figure 2.2: Credibility Distributions Φ1, Φ2 and Φ3


Theorem 2.20 Let ξ be a fuzzy variable with membership function µ. Thenits credibility distribution is

Φ(x) =12

(supy≤x

µ(y) + 1− supy>x

µ(y)), ∀x ∈ <. (2.40)

Proof: It follows from the credibility inversion theorem immediately.

Theorem 2.21 (Liu [135], Sufficient and Necessary Condition for Credibil-ity Distribution) A function Φ : < → [0, 1] is a credibility distribution if andonly if it is an increasing function with

limx→−∞

Φ(x) ≤ 0.5 ≤ limx→∞

Φ(x), (2.41)

limy↓x

Φ(y) = Φ(x) if limy↓x

Φ(y) > 0.5 or Φ(x) ≥ 0.5. (2.42)

Proof: It is obvious that a credibility distribution Φ is an increasing func-tion. The inequalities (2.41) follow from the credibility asymptotic theoremimmediately. Assume that x is a point at which limy↓x Φ(y) > 0.5. That is,

limy↓x

Crξ ≤ y > 0.5.

Since ξ ≤ y ↓ ξ ≤ x as y ↓ x, it follows from the credibility semicontinuitylaw that

Φ(y) = Crξ ≤ y ↓ Crξ ≤ x = Φ(x)

as y ↓ x. When x is a point at which Φ(x) ≥ 0.5, if limy↓x Φ(y) 6= Φ(x), thenwe have

limy↓x

Φ(y) > Φ(x) ≥ 0.5.

For this case, we have proved that limy↓x Φ(y) = Φ(x). Thus (2.41) and(2.42) are proved.

Conversely, if Φ : < → [0, 1] is an increasing function satisfying (2.41) and(2.42), then

µ(x) =

2Φ(x), if Φ(x) < 0.5

1, if limy↑x

Φ(y) < 0.5 ≤ Φ(x)

2− 2Φ(x), if 0.5 ≤ limy↑x

Φ(y)

(2.43)

takes values in [0, 1] and supµ(x) = 1. It follows from Theorem 2.17 thatthere is a fuzzy variable ξ whose membership function is just µ. Let us verifythat Φ is the credibility distribution of ξ, i.e., Crξ ≤ x = Φ(x) for each x.The argument breaks down into two cases. (i) If Φ(x) < 0.5, then we havesupy>x µ(y) = 1, and µ(y) = 2Φ(y) for each y with y ≤ x. Thus

Crξ ≤ x =12

(supy≤x

µ(y) + 1− supy>x

µ(y))

= supy≤x

Φ(y) = Φ(x).

Section 2.4 - Credibility Distribution 65

(ii) If Φ(x) ≥ 0.5, then we have supy≤x µ(y) = 1 and Φ(y) ≥ Φ(x) ≥ 0.5 foreach y with y > x. Thus µ(y) = 2− 2Φ(y) and

Crξ ≤ x =12

(supy≤x

µ(y) + 1− supy>x

µ(y))

=12

(1 + 1− sup

y>x(2− 2Φ(y))

)= inf

y>xΦ(y) = lim

y↓xΦ(y) = Φ(x).


Example 2.17: Let a and b be two numbers with 0 ≤ a ≤ 0.5 ≤ b ≤ 1. Wedefine a fuzzy variable by the following membership function,

µ(x) =

2a, if x < 01, if x = 0

2− 2b, if x > 0.

Then its credibility distribution is

Φ(x) =

a, if x < 0b, if x ≥ 0.

Thus we havelim

x→−∞Φ(x) = a, lim

x→+∞Φ(x) = b.

Theorem 2.22 A fuzzy variable with credibility distribution Φ is(a) nonnegative if and only if Φ(x) = 0 for all x < 0;(b) positive if and only if Φ(x) = 0 for all x ≤ 0.

Proof: It follows immediately from the definition.

Theorem 2.23 Let ξ be a fuzzy variable. Then we have(a) if ξ is simple, then its credibility distribution is a simple function;(b) if ξ is discrete, then its credibility distribution is a step function;(c) if ξ is continuous on the real line <, then its credibility distribution is acontinuous function.

Proof: The parts (a) and (b) follow immediately from the definition. Thepart (c) follows from Theorem 2.20 and the continuity of the membershipfunction.

Example 2.18: However, the inverse of Theorem 2.23 is not true. Forexample, let ξ be a fuzzy variable whose membership function is

µ(x) =

x, if 0 ≤ x ≤ 11, otherwise.


Then its credibility distribution is Φ(x) ≡ 0.5. It is clear that Φ(x) is simpleand continuous. But the fuzzy variable ξ is neither simple nor continuous.

Definition 2.13 A continuous fuzzy variable is said to be (a) singular if itscredibility distribution is a singular function; (b) absolutely continuous if itscredibility distribution is absolutely continuous.

Definition 2.14 (Liu [130]) The credibility density function φ: < → [0,+∞)of a fuzzy variable ξ is a function such that

Φ(x) =∫ x

−∞φ(y)dy, ∀x ∈ <, (2.44)

∫ +∞

−∞φ(y)dy = 1 (2.45)

where Φ is the credibility distribution of the fuzzy variable ξ.

Example 2.19: The credibility density function of a triangular fuzzy vari-able (a, b, c) is

φ(x) =

1

2(b− a), if a ≤ x ≤ b

12(c− b)

, if b ≤ x ≤ c

0, otherwise.

Example 2.20: The credibility density function of a trapezoidal fuzzy vari-able (a, b, c, d) is

φ(x) =

1

2(b− a), if a ≤ x ≤ b

12(d− c)

, if c ≤ x ≤ d

0, otherwise.

Example 2.21: The credibility density function of an equipossible fuzzyvariable (a, b) does not exist.

Example 2.22: The credibility density function does not necessarily existeven if the membership function is continuous and unimodal with a finitesupport. Let f be the Cantor function, and set

µ(x) =

f(x), if 0 ≤ x ≤ 1

f(2− x), if 1 < x ≤ 20, otherwise.

(2.46)


Then µ is a continuous and unimodal function with µ(1) = 1. Hence µis a membership function. However, its credibility distribution is not anabsolutely continuous function. Thus the credibility density function doesnot exist.

Theorem 2.24 Let ξ be a fuzzy variable whose credibility density functionφ exists. Then we have

Crξ ≤ x =∫ x

−∞φ(y)dy, Crξ ≥ x =

∫ +∞

x

φ(y)dy. (2.47)

Proof: The first part follows immediately from the definition. In addition,by the self-duality of credibility measure, we have

Crξ ≥ x = 1− Crξ < x =∫ +∞

−∞φ(y)dy −

∫ x

−∞φ(y)dy =

∫ +∞

x

φ(y)dy.


Example 2.23: Different from the random case, generally speaking,

Cra ≤ ξ ≤ b 6=∫ b

a

φ(y)dy.

Consider the trapezoidal fuzzy variable ξ = (1, 2, 3, 4). Then Cr2 ≤ ξ ≤3 = 0.5. However, it is obvious that φ(x) = 0 when 2 ≤ x ≤ 3 and∫ 3

2

φ(y)dy = 0 6= 0.5 = Cr2 ≤ ξ ≤ 3.

Joint Credibility Distribution

Definition 2.15 Let (ξ1, ξ2, · · · , ξn) be a fuzzy vector. Then the joint credi-bility distribution Φ : <n → [0, 1] is defined by

Φ(x1, x2, · · · , xn) = Crθ ∈ Θ

∣∣ ξ1(θ) ≤ x1, ξ2(θ) ≤ x2, · · · , ξn(θ) ≤ xn

.

Definition 2.16 The joint credibility density function φ : <n → [0,+∞) ofa fuzzy vector (ξ1, ξ2, · · · , ξn) is a function such that

Φ(x1, x2, · · · , xn) =∫ x1

−∞

∫ x2

−∞· · ·∫ xn

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn

holds for all (x1, x2, · · · , xn) ∈ <n, and∫ +∞

−∞

∫ +∞

−∞· · ·∫ +∞

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn = 1

where Φ is the joint credibility distribution of the fuzzy vector (ξ1, ξ2, · · · , ξn).


2.5 Independence

The independence of fuzzy variables has been discussed by many authorsfrom different angles. Here we use the following condition.

Definition 2.17 (Liu and Gao [158]) The fuzzy variables ξ1, ξ2, · · · , ξm aresaid to be independent if

Cr

m⋂

i=1

ξi ∈ Bi

= min

1≤i≤mCr ξi ∈ Bi (2.48)

for any sets B1, B2, · · · , Bm of <.

Theorem 2.25 The fuzzy variables ξ1, ξ2, · · · , ξm are independent if andonly if

Cr

m⋃

i=1

ξi ∈ Bi

= max

1≤i≤mCr ξi ∈ Bi (2.49)

for any sets B1, B2, · · · , Bm of <.

Proof: It follows from the self-duality of credibility measure that ξ1, ξ2, · · · , ξmare independent if and only if

Cr

m⋃

i=1

ξi ∈ Bi

= 1− Cr

m⋂

i=1

ξi ∈ Bci

= 1− min

1≤i≤mCrξi ∈ Bc

i = max1≤i≤m

Cr ξi ∈ Bi .

Thus (2.49) is verified. The proof is complete.

Theorem 2.26 The fuzzy variables ξ1, ξ2, · · · , ξm are independent if andonly if

Cr

m⋂

i=1

ξi = xi

= min

1≤i≤mCr ξi = xi (2.50)

for any real numbers x1, x2, · · · , xm.

Proof: If ξ1, ξ2, · · · , ξm are independent, then we have (2.50) immediatelyby taking Bi = xi for each i. Conversely, if Cr∩m

i=1(ξi ∈ Bi) ≥ 0.5, itfollows from Theorem 2.2 that (2.48) holds. Otherwise, we have Cr∩m

i=1(ξi =xi) < 0.5 for any real numbers xi ∈ Bi, i = 1, 2, · · · ,m, and

Cr

m⋂

i=1

ξi ∈ Bi

= Cr

⋃xi∈Bi,1≤i≤m

m⋂i=1

ξi = xi

= sup

xi∈Bi,1≤i≤mCr

m⋂

i=1

ξi = xi

= sup

xi∈Bi,1≤i≤mmin

1≤i≤mCrξi = xi

= min1≤i≤m

supxi∈Bi

Cr ξi = xi = min1≤i≤m

Cr ξi ∈ Bi .


Hence (2.48) is true, and ξ1, ξ2, · · · , ξm are independent. The theorem is thusproved.

Theorem 2.27 Let µi be membership functions of fuzzy variables ξi, i =1, 2, · · · ,m, respectively, and µ the joint membership function of fuzzy vector(ξ1, ξ2, · · · , ξm). Then the fuzzy variables ξ1, ξ2, · · · , ξm are independent ifand only if

µ(x1, x2, · · · , xm) = min1≤i≤m

µi(xi) (2.51)


Proof: Suppose that ξ1, ξ2, · · · , ξm are independent. It follows from Theo-rem 2.26 that

µ(x1, x2, · · · , xm) =

(2Cr

m⋂

i=1

ξi = xi

)∧ 1

=(

2 min1≤i≤m

Crξi = xi)∧ 1

= min1≤i≤m

(2Crξi = xi) ∧ 1 = min1≤i≤m

µi(xi).

Conversely, for any real numbers x1, x2, · · · , xm with Cr∩mi=1ξi = xi <

0.5, we have

Cr

m⋂

i=1

ξi = xi

=

12

(2Cr

m⋂

i=1

ξi = xi

)∧ 1

=12µ(x1, x2, · · · , xm) =

12

min1≤i≤m

µi(xi)

=12

(min

1≤i≤m(2Cr ξi = xi) ∧ 1

)= min

1≤i≤mCr ξi = xi .

It follows from Theorem 2.26 that ξ1, ξ2, · · · , ξm are independent. The theo-rem is proved.

Theorem 2.28 Let Φi be credibility distributions of fuzzy variables ξi, i =1, 2, · · · ,m, respectively, and Φ the joint credibility distribution of fuzzy vector(ξ1, ξ2, · · · , ξm). If ξ1, ξ2, · · · , ξm are independent, then we have

Φ(x1, x2, · · · , xm) = min1≤i≤m

Φi(xi) (2.52)



Proof: Since ξ1, ξ2, · · · , ξm are independent fuzzy variables, we have

Φ(x1, x2, · · · , xm) = Cr

m⋂

i=1

ξi ≤ xi

= min

1≤i≤mCrξi ≤ xi = min

1≤i≤mΦi(xi)

for any real numbers x1, x2, · · · , xm. The theorem is proved.

Example 2.24: However, the equation (2.52) does not imply that the fuzzyvariables are independent. For example, let ξ be a fuzzy variable with credi-bility distribution Φ. Then the joint credibility distribution Ψ of fuzzy vector(ξ, ξ) is

Ψ(x1, x2) = Crξ ≤ x1, ξ ≤ x2 = Crξ ≤ x1 ∧Crξ ≤ x2 = Φ(x1)∧Φ(x2)

for any real numbers x1 and x2. But, generally speaking, a fuzzy variable isnot independent with itself.

Theorem 2.29 Let ξ1, ξ2, · · · , ξm be independent fuzzy variables, and f1, f2,· · · , fn are real-valued functions. Then f1(ξ1), f2(ξ2), · · · , fm(ξm) are inde-pendent fuzzy variables.

Proof: For any sets B1, B2, · · · , Bm of <, we have

Cr

m⋂

i=1

fi(ξi) ∈ Bi

= Cr

m⋂

i=1

ξi ∈ f−1i (Bi)

= min

1≤i≤mCrξi ∈ f−1

i (Bi) = min1≤i≤m

Crfi(ξi) ∈ Bi.

Thus f1(ξ1), f2(ξ2), · · · , fm(ξm) are independent fuzzy variables.


Definition 2.18 (Liu [135]) The fuzzy variables ξ and η are said to be iden-tically distributed if

Crξ ∈ B = Crη ∈ B (2.53)

for any set B of <.

Theorem 2.30 The fuzzy variables ξ and η are identically distributed if andonly if ξ and η have the same membership function.

Proof: Let µ and ν be the membership functions of ξ and η, respectively. Ifξ and η are identically distributed fuzzy variables, then, for any x ∈ <, wehave

µ(x) = (2Crξ = x) ∧ 1 = (2Crη = x) ∧ 1 = ν(x).

Thus ξ and η have the same membership function.

Section 2.7 - Extension Principle of Zadeh 71

Conversely, if ξ and η have the same membership function, i.e., µ(x) ≡ν(x), then, by using the credibility inversion theorem, we have

Crξ ∈ B =12

(supx∈B


µ(x))

=12

(supx∈B

ν(x) + 1− supx∈Bc

ν(x))

= Crη ∈ B

for any set B of <. Thus ξ and η are identically distributed fuzzy variables.

Theorem 2.31 The fuzzy variables ξ and η are identically distributed if andonly if Crξ = x = Crη = x for each x ∈ <.

Proof: If ξ and η are identically distributed fuzzy variables, then we im-mediately have Crξ = x = Crη = x for each x. Conversely, it followsfrom

µ(x) = (2Crξ = x) ∧ 1 = (2Crη = x) ∧ 1 = ν(x)

that ξ and η have the same membership function. Thus ξ and η are identicallydistributed fuzzy variables.

Theorem 2.32 If ξ and η are identically distributed fuzzy variables, then ξand η have the same credibility distribution.

Proof: If ξ and η are identically distributed fuzzy variables, then, for anyx ∈ <, we have Crξ ∈ (−∞, x] = Crη ∈ (−∞, x]. Thus ξ and η have thesame credibility distribution.

Example 2.25: The inverse of Theorem 2.32 is not true. We consider twofuzzy variables with the following membership functions,

µ(x) =

1.0, if x = 00.6, if x = 10.8, if x = 2,

ν(x) =

1.0, if x = 00.7, if x = 10.8, if x = 2.

It is easy to verify that ξ and η have the same credibility distribution,

Φ(x) =

0, if x < 0

0.6, if 0 ≤ x < 21, if x ≥ 2.

However, they are not identically distributed fuzzy variables.

Theorem 2.33 Let ξ and η be two fuzzy variables whose credibility densityfunctions exist. If ξ and η are identically distributed, then they have the samecredibility density function.

Proof: It follows from Theorem 2.32 that the fuzzy variables ξ and η have thesame credibility distribution. Hence they have the same credibility densityfunction.


2.7 Extension Principle of Zadeh

Theorem 2.34 (Extension Principle of Zadeh) Let ξ1, ξ2, · · · , ξn be inde-pendent fuzzy variables with membership functions µ1, µ2, · · · , µn, respec-tively, and f : <n → < a function. Then the membership function µ ofξ = f(ξ1, ξ2, · · · , ξn) is derived from the membership functions µ1, µ2, · · · , µn

byµ(x) = sup

x=f(x1,x2,··· ,xn)

min1≤i≤n

µi(xi) (2.54)

for any x ∈ <. Here we set µ(x) = 0 if there are not real numbers x1, x2, · · · , xn

such that x = f(x1, x2, · · · , xn).

Proof: It follows from Definition 2.10 that the membership function of ξ =f(ξ1, ξ2, · · · , ξn) is

µ(x) = (2Cr f(ξ1, ξ2, · · · , ξn) = x) ∧ 1

=

2Cr

⋃x=f(x1,x2,··· ,xn)

ξ1 = x1, ξ2 = x2, · · · , ξn = xn

∧ 1

=

(2 sup

x=f(x1,x2,··· ,xn)

Crξ1 = x1, ξ2 = x2, · · · , ξn = xn

)∧ 1

=

(2 sup

x=f(x1,x2,··· ,xn)

min1≤k≤n

Crξi = xi

)∧ 1 (by independence)

= supx=f(x1,x2,··· ,xn)

min1≤k≤n

(2Crξi = xi) ∧ 1

= supx=f(x1,x2,··· ,xn)

min1≤i≤n

µi(xi).


Remark 2.5: The extension principle of Zadeh is only applicable to theoperations on independent fuzzy variables. In the past literature, the exten-sion principle is used as a postulate. However, it is treated as a theorem incredibility theory.

Example 2.26: The sum of independent equipossible fuzzy variables ξ =(a1, a2) and η = (b1, b2) is also an equipossible fuzzy variable, and

ξ + η = (a1 + b1, a2 + b2).

Their product is also an equipossible fuzzy variable, and

ξ · η =(

mina1≤x≤a2,b1≤y≤b2

xy, maxa1≤x≤a2,b1≤y≤b2

xy

).


Example 2.27: The sum of independent triangular fuzzy variables ξ =(a1, a2, a3) and η = (b1, b2, b3) is also a triangular fuzzy variable, and

ξ + η = (a1 + b1, a2 + b2, a3 + b3).

The product of a triangular fuzzy variable ξ = (a1, a2, a3) and a scalar numberλ is

λ · ξ =

(λa1, λa2, λa3), if λ ≥ 0

(λa3, λa2, λa1), if λ < 0.

That is, the product of a triangular fuzzy variable and a scalar number isalso a triangular fuzzy variable. However, the product of two triangular fuzzyvariables is not a triangular one.

Example 2.28: The sum of independent trapezoidal fuzzy variables ξ =(a1, a2, a3, a4) and η = (b1, b2, b3, b4) is also a trapezoidal fuzzy variable, and

ξ + η = (a1 + b1, a2 + b2, a3 + b3, a4 + b4).

The product of a trapezoidal fuzzy variable ξ = (a1, a2, a3, a4) and a scalarnumber λ is

λ · ξ =

(λa1, λa2, λa3, λa4), if λ ≥ 0

(λa4, λa3, λa2, λa1), if λ < 0.

That is, the product of a trapezoidal fuzzy variable and a scalar number isalso a trapezoidal fuzzy variable. However, the product of two trapezoidalfuzzy variables is not a trapezoidal one.

Example 2.29: Let ξ1, ξ2, · · · , ξn be independent fuzzy variables with mem-bership functions µ1, µ2, · · · , µn, respectively, and f : <n → < a function.Then for any set B of real numbers, the credibility Crf(ξ1, ξ2, · · · , ξn) ∈ Bis

12

(sup

f(x1,x2,··· ,xn)∈B

min1≤i≤n

µi(xi) + 1− supf(x1,x2,··· ,xn)∈Bc

min1≤i≤n

µi(xi)

).

2.8 Expected Value

There are many ways to define an expected value operator for fuzzy variables.The most general definition of expected value operator of fuzzy variable wasgiven by Liu and Liu [132]. This definition is applicable to not only continuousfuzzy variables but also discrete ones.

Definition 2.19 (Liu and Liu [132]) Let ξ be a fuzzy variable. Then theexpected value of ξ is defined by

E[ξ] =∫ +∞

0

Crξ ≥ rdr −∫ 0

−∞Crξ ≤ rdr (2.55)



Example 2.30: Let ξ be the equipossible fuzzy variable (a, b). If a ≥ 0,then Crξ ≤ r ≡ 0 when r < 0, and

Crξ ≥ r =

1, if r ≤ a

0.5, if a < r ≤ b

0, if r > b,

E[ξ] =

(∫ a

0

1dr +∫ b

a

0.5dr +∫ +∞

b

0dr

)−∫ 0

−∞0dr =

a+ b

2.

If b ≤ 0, then Crξ ≥ r ≡ 0 when r > 0, and

Crξ ≤ r =

1, if r ≥ b

0.5, if a ≤ r < b

0, if r < a,

E[ξ] =∫ +∞

0

0dr −

(∫ a

−∞0dr +

∫ b

a

0.5dr +∫ 0

b

1dr

)=a+ b

2.

If a < 0 < b, then

Crξ ≥ r =

0.5, if 0 ≤ r ≤ b

0, if r > b,

Crξ ≤ r =

0, if r < a

0.5, if a ≤ r ≤ 0,

E[ξ] =

(∫ b

0

0.5dr +∫ +∞

b

0dr

)−(∫ a

−∞0dr +

∫ 0

a

0.5dr)

=a+ b

2.

Thus we always have the expected value (a+ b)/2.

Example 2.31: The triangular fuzzy variable ξ = (a, b, c) has an expectedvalue E[ξ] = (a+ 2b+ c)/4.

Example 2.32: The trapezoidal fuzzy variable ξ = (a, b, c, d) has an ex-pected value E[ξ] = (a+ b+ c+ d)/4.

Example 2.33: Let ξ be a continuous fuzzy variable with membership func-tion µ. If its expected value exists, and there is a point x0 such that µ(x) is in-creasing on (−∞, x0) and decreasing on (x0,+∞), then Crξ ≥ x = µ(x)/2for any x > x0 and Crξ ≤ x = µ(x)/2 for any x < x0. Thus

E[ξ] = x0 +12

∫ +∞

x0

µ(x)dx− 12

∫ x0

−∞µ(x)dx.


Example 2.34: Let ξ be a fuzzy variable with membership function

µ(x) =

0, if x < 0x, if 0 ≤ x ≤ 11, if x > 1.

Then its expected value is +∞. If ξ is a fuzzy variable with membershipfunction

µ(x) =

1, if x < 0

1− x, if 0 ≤ x ≤ 10, if x > 1.

Then its expected value is −∞.

Example 2.35: The definition of expected value operator is also applicableto discrete case. Assume that ξ is a simple fuzzy variable whose membershipfunction is given by

µ(x) =

µ1, if x = x1

µ2, if x = x2

· · ·µm, if x = xm

(2.56)

where x1, x2, · · · , xm are distinct numbers. Note that µ1 ∨µ2 ∨ · · · ∨µm = 1.Definition 2.19 implies that the expected value of ξ is

E[ξ] =m∑

i=1

wixi (2.57)

where the weights are given by

wi =12

(max

1≤j≤mµj |xj ≤ xi − max

1≤j≤mµj |xj < xi

+ max1≤j≤m

µj |xj ≥ xi − max1≤j≤m

µj |xj > xi)

for i = 1, 2, · · · ,m. It is easy to verify that all wi ≥ 0 and the sum of allweights is just 1.

Example 2.36: Consider the fuzzy variable ξ defined by (2.56). Supposex1 < x2 < · · · < xm and there exists an index k with 1 < k < m such that

µ1 ≤ µ2 ≤ · · · ≤ µk and µk ≥ µk+1 ≥ · · · ≥ µm.


Note that µk ≡ 1. Then the expected value is determined by (2.57) and theweights are given by

wi =

µ1

2, if i = 1

µi − µi−1

2, if i = 2, 3, · · · , k − 1

1− µk−1 + µk+1

2, if i = k

µi − µi+1

2, if i = k + 1, k + 2, · · · ,m− 1

µm

2, if i = m.

Theorem 2.35 (Liu [130]) Let ξ be a fuzzy variable whose credibility densityfunction φ exists. If the Lebesgue integral∫ +∞

−∞xφ(x)dx


E[ξ] =∫ +∞

−∞xφ(x)dx. (2.58)

Proof: It follows from the definition of expected value operator and FubiniTheorem that

E[ξ] =∫ +∞

0


−∞Crξ ≤ rdr

=∫ +∞

0

[∫ +∞

r

φ(x)dx]

dr −∫ 0

−∞

[∫ r

−∞φ(x)dx

]dr

=∫ +∞

0

[∫ x

0

φ(x)dr]

dx−∫ 0

−∞

[∫ 0

x

φ(x)dr]

dx

=∫ +∞

0

xφ(x)dx+∫ 0

−∞xφ(x)dx

=∫ +∞

−∞xφ(x)dx.


Example 2.37: Let ξ be a fuzzy variable with credibility distribution Φ.Generally speaking,

E[ξ] 6=∫ +∞

−∞xdΦ(x).


For example, let ξ be a fuzzy variable with membership function

µ(x) =

0, if x < 0x, if 0 ≤ x ≤ 11, if x > 1.

Then E[ξ] = +∞. However∫ +∞

−∞xdΦ(x) =

146= +∞.

Theorem 2.36 (Liu [135]) Let ξ be a fuzzy variable with credibility distri-bution Φ. If

limx→−∞

Φ(x) = 0, limx→∞

Φ(x) = 1

and the Lebesgue-Stieltjes integral∫ +∞

−∞xdΦ(x)


E[ξ] =∫ +∞

−∞xdΦ(x). (2.59)


diately have

limy→+∞

∫ y

0

xdΦ(x) =∫ +∞

0


∫ 0

y

xdΦ(x) =∫ 0

−∞xdΦ(x)

and

limy→+∞

∫ +∞

y


∫ y

−∞xdΦ(x) = 0.


y

xdΦ(x) ≥ y

(lim

z→+∞Φ(z)− Φ(y)

)= y (1− Φ(y)) ≥ 0, for y > 0,

∫ y

−∞xdΦ(x) ≤ y

(Φ(y)− lim

z→−∞Φ(z)

)= yΦ(y) ≤ 0, for y < 0

thatlim

y→+∞y (1− Φ(y)) = 0, lim

y→−∞yΦ(y) = 0.


n−1∑i=0

xi (Φ(xi+1)− Φ(xi)) →∫ y

0

xdΦ(x)


andn−1∑i=0

(1− Φ(xi+1))(xi+1 − xi) →∫ y

0

Crξ ≥ rdr


n−1∑i=0

xi (Φ(xi+1)− Φ(xi))−n−1∑i=0

(1− Φ(xi+1)(xi+1 − xi) = y(Φ(y)− 1) → 0


0

Crξ ≥ rdr =∫ +∞

0

xdΦ(x).


−∫ 0

−∞Crξ ≤ rdr =

∫ 0

−∞xdΦ(x).

It follows that the equation (2.59) holds.

Linearity of Expected Value Operator

Theorem 2.37 (Liu and Liu [151]) Let ξ and η be independent fuzzy vari-ables with finite expected values. Then for any numbers a and b, we have

E[aξ + bη] = aE[ξ] + bE[η]. (2.60)

Proof: Step 1: We first prove that E[ξ + b] = E[ξ] + b for any real numberb. If b ≥ 0, we have

E[ξ + b] =∫ ∞

0

Crξ + b ≥ rdr −∫ 0

−∞Crξ + b ≤ rdr

=∫ ∞

0

Crξ ≥ r − bdr −∫ 0

−∞Crξ ≤ r − bdr

= E[ξ] +∫ b

0

(Crξ ≥ r − b+ Crξ < r − b) dr

= E[ξ] + b.


E[ξ + b] = E[ξ]−∫ 0

b

(Crξ ≥ r − b+ Crξ < r − b) dr = E[ξ] + b.


Step 2: We prove that E[aξ] = aE[ξ] for any real number a. If a = 0,then the equation E[aξ] = aE[ξ] holds trivially. If a > 0, we have

E[aξ] =∫ ∞

0

Craξ ≥ rdr −∫ 0

−∞Craξ ≤ rdr

=∫ ∞

0

Crξ ≥ r

a

dr −

∫ 0

−∞Crξ ≤ r

a

dr

= a

∫ ∞

0

Crξ ≥ r

a

d( ra

)− a

∫ 0

−∞Crξ ≤ r

a

d( ra

)= aE[ξ].

If a < 0, we have

E[aξ] =∫ ∞

0

Craξ ≥ rdr −∫ 0

−∞Craξ ≤ rdr

=∫ ∞

0

Crξ ≤ r

a

dr −

∫ 0

−∞Crξ ≥ r

a

dr

= a

∫ ∞

0

Crξ ≥ r

a

d( ra

)− a

∫ 0

−∞Crξ ≤ r

a

d( ra

)= aE[ξ].

Step 3: We prove that E[ξ + η] = E[ξ] + E[η] when both ξ and η aresimple fuzzy variables with the following membership functions,

µ(x) =

µ1, if x = a1

µ2, if x = a2

· · ·µm, if x = am,

ν(x) =

ν1, if x = b1

ν2, if x = b2

· · ·νn, if x = bn.

Then ξ+η is also a simple fuzzy variable taking values ai+bj with membershipdegrees µi ∧ νj , i = 1, 2, · · · ,m, j = 1, 2, · · · , n, respectively. Now we define

w′i =12

(max

1≤k≤mµk|ak ≤ ai − max

1≤k≤mµk|ak < ai

+ max1≤k≤m

µk|ak ≥ ai − max1≤k≤m

µk|ak > ai),

w′′j =12

(max

1≤l≤nνl|bl ≤ bi − max

1≤l≤nνl|bl < bi

+ max1≤l≤n

νl|bl ≥ bi − max1≤l≤n

νl|bl > bi),


wij =12

(max

1≤k≤m,1≤l≤nµk ∧ νl|ak + bl ≤ ai + bj

− max1≤k≤m,1≤l≤n

µk ∧ νl|ak + bl < ai + bj

+ max1≤k≤m,1≤l≤n

µk ∧ νl|ak + bl ≥ ai + bj

− max1≤k≤m,1≤l≤n

µk ∧ νl|ak + bl > ai + bj)

for i = 1, 2, · · · ,m and j = 1, 2, · · · , n. It is also easy to verify that

w′i =n∑

j=1

wij , w′′j =m∑

i=1

wij

for i = 1, 2, · · · ,m and j = 1, 2, · · · , n. If ai, bj and ai + bj aresequences consisting of distinct elements, then

E[ξ] =m∑

i=1

aiw′i, E[η] =

n∑j=1

bjw′′j , E[ξ + η] =

m∑i=1

n∑j=1

(ai + bj)wij .

Thus E[ξ + η] = E[ξ] +E[η]. If not, we may give them a small perturbationsuch that they are distinct, and prove the linearity by letting the perturbationtend to zero.

Step 4: We prove that E[ξ + η] = E[ξ] + E[η] when ξ and η are fuzzyvariables such that

limy↑0

Crξ ≤ y ≤ 12≤ Crξ ≤ 0,

limy↑0

Crη ≤ y ≤ 12≤ Crη ≤ 0.

(2.61)

We define simple fuzzy variables ξi via credibility distributions as follows,

Φi(x) =

k − 12i

, ifk − 1

2i≤ Crξ ≤ x < k

2i, k = 1, 2, · · · , 2i−1

k

2i, if

k − 12i

≤ Crξ ≤ x < k

2i, k = 2i−1 + 1, · · · , 2i

1, if Crξ ≤ x = 1

for i = 1, 2, · · · Thus ξi is a sequence of simple fuzzy variables satisfying

Crξi ≤ r ↑ Crξ ≤ r, if r ≤ 0Crξi ≥ r ↑ Crξ ≥ r, if r ≥ 0


as i → ∞. Similarly, we define simple fuzzy variables ηi via credibilitydistributions as follows,

Ψi(x) =

k − 12i

, ifk − 1

2i≤ Crη ≤ x < k

2i, k = 1, 2, · · · , 2i−1

k

2i, if

k − 12i

≤ Crη ≤ x < k

2i, k = 2i−1 + 1, · · · , 2i

1, if Crη ≤ x = 1

for i = 1, 2, · · · Thus ηi is a sequence of simple fuzzy variables satisfying

Crηi ≤ r ↑ Crη ≤ r, if r ≤ 0Crηi ≥ r ↑ Crη ≥ r, if r ≥ 0

as i→∞. It is also clear that ξi+ηi is a sequence of simple fuzzy variables.Furthermore, when r ≤ 0, it follows from (2.61) that

limi→∞

Crξi + ηi ≤ r = limi→∞

supx≤0,y≤0,x+y≤r

Crξi ≤ x ∧ Crηi ≤ y

= supx≤0,y≤0,x+y≤r

limi→∞

Crξi ≤ x ∧ Crηi ≤ y

= supx≤0,y≤0,x+y≤r

Crξ ≤ x ∧ Crη ≤ y

= Crξ + η ≤ r.

That is,Crξi + ηi ≤ r ↑ Crξ + η ≤ r, if r ≤ 0.


Crξi + ηi ≥ r ↑ Crξ + η ≥ r, if r ≥ 0.

Since the expected values E[ξ] and E[η] exist, we have

E[ξi] =∫ +∞

0

Crξi ≥ rdr −∫ 0

−∞Crξi ≤ rdr

→∫ +∞

0


−∞Crξ ≤ rdr = E[ξ],

E[ηi] =∫ +∞

0

Crηi ≥ rdr −∫ 0

−∞Crηi ≤ rdr

→∫ +∞

0

Crη ≥ rdr −∫ 0

−∞Crη ≤ rdr = E[η],


E[ξi + ηi] =∫ +∞

0

Crξi + ηi ≥ rdr −∫ 0

−∞Crξi + ηi ≤ rdr

→∫ +∞

0

Crξ + η ≥ rdr −∫ 0

−∞Crξ + η ≤ rdr = E[ξ + η]

as i→∞. It follows from Step 3 that E[ξ + η] = E[ξ] + E[η].

Step 5: We prove that E[ξ + η] = E[ξ] + E[η] when ξ and η are arbi-trary fuzzy variables. Since they have finite expected values, there exist twonumbers c and d such that

limy↑0

Crξ + c ≤ y ≤ 12≤ Crξ + c ≤ 0,

limy↑0

Crη + d ≤ y ≤ 12≤ Crη + d ≤ 0.

It follows from Steps 1 and 4 that

E[ξ + η] = E[(ξ + c) + (η + d)− c− d]

= E[(ξ + c) + (η + d)]− c− d

= E[ξ + c] + E[η + d]− c− d

= E[ξ] + c+ E[η] + d− c− d

= E[ξ] + E[η].

Step 6: We prove that E[aξ + bη] = aE[ξ] + bE[η] for any real numbersa and b. In fact, the equation follows immediately from Steps 2 and 5. Thetheorem is proved.

Example 2.38: Theorem 2.37 does not hold if ξ and η are not independent.For example, take (Θ,P,Cr) to be θ1, θ2, θ3 with Crθ1 = 0.7, Crθ2 =0.3 and Crθ3 = 0.2. The fuzzy variables are defined by

ξ1(θ) =

1, if θ = θ1

0, if θ = θ2

2, if θ = θ3,

ξ2(θ) =

0, if θ = θ1

2, if θ = θ2

3, if θ = θ3.

Then we have

(ξ1 + ξ2)(θ) =

1, if θ = θ1

2, if θ = θ2

5, if θ = θ3.

Thus E[ξ1] = 0.9, E[ξ2] = 0.8, and E[ξ1 + ξ2] = 1.9. This fact implies that

E[ξ1 + ξ2] > E[ξ1] + E[ξ2].


If the fuzzy variables are defined by

η1(θ) =

0, if θ = θ1

1, if θ = θ2

2, if θ = θ3,

η2(θ) =

0, if θ = θ1

3, if θ = θ2

1, if θ = θ3.

Then we have

(η1 + η2)(θ) =

0, if θ = θ1

4, if θ = θ2

3, if θ = θ3.

Thus E[η1] = 0.5, E[η2] = 0.9, and E[η1 + η2] = 1.2. This fact implies that

E[η1 + η2] < E[η1] + E[η2].

Expected Value of Function of Fuzzy Variable

Let ξ be a fuzzy variable, and f : < → < a function. Then the expectedvalue of f(ξ) is

E[f(ξ)] =∫ +∞

0

Crf(ξ) ≥ rdr −∫ 0

−∞Crf(ξ) ≤ rdr.

For random case, it has been proved that the expected value E[f(ξ)] is theLebesgue-Stieltjes integral of f(x) with respect to the probability distributionΦ of ξ if the integral exists. However, generally speaking, it is not true forfuzzy case.

Example 2.39: We consider a fuzzy variable ξ whose membership functionis given by

µ(x) =

0.6, if − 1 ≤ x < 01, if 0 ≤ x ≤ 10, otherwise.

Then the expected value E[ξ2] = 0.5. However, the credibility distributionof ξ is

Φ(x) =

0, if x < −1

0.3, if − 1 ≤ x < 00.5, if 0 ≤ x < 11, if x ≥ 1


−∞x2dΦ(x) = (−1)2 × 0.3 + 02 × 0.2 + 12 × 0.5 = 0.8 6= E[ξ2].


Theorem 2.38 (Zhu and Ji [276]) Let ξ be a fuzzy variable whose credibilitydistribution Φ satisfies

limx→−∞

Φ(x) = 0, limx→∞

Φ(x) = 1. (2.62)

If f(x) is a monotone function such that the Lebesgue-Stieltjes integral∫ +∞

−∞f(x)dΦ(x) (2.63)


E[f(ξ)] =∫ +∞

−∞f(x)dΦ(x). (2.64)

Proof: We first suppose that f(x) is a monotone increasing function. Sincethe Lebesgue-Stieltjes integral (2.63) is finite, we immediately have

limy→+∞

Crξ ≥ yf(y) = limy→+∞

(1− Φ(y))f(y) = 0, (2.65)

limy→−∞

Crξ ≤ yf(y) = limy→−∞

Φ(y)f(y) = 0. (2.66)

Assume that a and b are two real numbers such that a < 0 < b. Theintegration by parts produces∫ b

0

Crf(ξ) ≥ rdr =∫ b

0

Crξ ≥ f−1(r)dr =∫ f−1(b)

f−1(0)

Crξ ≥ ydf(y)

= Crξ ≥ f−1(b)f(f−1(b))−∫ f−1(b)

f−1(0)

f(y)dCrξ ≥ y

= Crξ ≥ f−1(b)f(f−1(b)) +∫ f−1(b)

f−1(0)

f(y)dΦ(y).

Using (2.65) and letting b→ +∞, we obtain∫ +∞

0

Crf(ξ) ≥ rdr =∫ +∞

f−1(0)

f(y)dΦ(y). (2.67)

In addition,∫ 0

a

Crf(ξ) ≤ rdr =∫ 0

a

Crξ ≤ f−1(r)dr =∫ f−1(0)

f−1(a)

Crξ ≤ ydf(y)

= −Crξ ≤ f−1(a)f(f−1(a))−∫ f−1(0)

f−1(a)

f(y)dCrξ ≤ y

= −Crξ ≤ f−1(a)f(f−1(a))−∫ f−1(0)

f−1(a)

f(y)dΦ(y).


Using (2.66) and letting a→ −∞, we obtain

∫ 0

−∞Crf(ξ) ≤ rdr = −

∫ f−1(0)

−∞f(y)dΦ(y). (2.68)


E[f(ξ)] =∫ +∞

0

Crf(ξ) ≥ r −∫ 0

−∞Crf(ξ) ≤ rdr =

∫ +∞

−∞f(y)dΦ(y).

If f(x) is a monotone decreasing function, then −f(x) is a monotone increas-ing function. Hence

E[f(ξ)] = −E[−f(ξ)] = −∫ +∞

−∞−f(x)dΦ(x) =

∫ +∞

−∞f(y)dΦ(y).

The theorem is verified.

Sum of a Fuzzy Number of Fuzzy Variables

Theorem 2.39 (Zhao and Liu [265]) Assume that ξi is a sequence of iidfuzzy variables, and n is a positive fuzzy integer (i.e., a fuzzy variable taking“positive integer” values) that is independent of the sequence ξi. Then wehave

E

[n∑

i=1

ξi

]= E [nξ1] . (2.69)

Proof: Since ξi is a sequence of iid fuzzy variables and n is independentof ξi, we have

Cr

n∑i=1

ξi ≥ r

= sup

n,x1+x2+···+xn≥rCrn = n ∧

(min

1≤i≤nCr ξi = xi

)≥ sup

nx≥rCrn = n ∧

(min

1≤i≤nCr ξi = x

)= sup

nx≥rCrn = n ∧ Crξ1 = x

= Cr nξ1 ≥ r .

On the other hand, for any given ε > 0, there exists an integer n and realnumbers x1, x2, · · · , xn with x1 + x2 + · · ·+ xn ≥ r such that

Cr

n∑

i=1

ξi ≥ r

− ε ≤ Crn = n ∧ Crξi = xi


for each i with 1 ≤ i ≤ n. Without loss of generality, we assume that nx1 ≥ r.Then we have

Cr

n∑

i=1

ξi ≥ r

− ε ≤ Crn = n ∧ Crξ1 = x1 ≤ Cr nξ1 ≥ r .

Letting ε→ 0, we get

Cr

n∑

i=1

ξi ≥ r

≤ Cr nξ1 ≥ r .

It follows that

Cr

n∑

i=1

ξi ≥ r

= Cr nξ1 ≥ r .

Similarly, the above identity still holds if the symbol “≥” is replaced with“≤”. Finally, by the definition of expected value operator, we have

E

[n∑

i=1

ξi

]=∫ +∞

0

Cr

n∑

i=1

ξi ≥ r

dr −

∫ 0

−∞Cr

n∑

i=1

ξi ≤ r

dr

=∫ +∞

0

Cr nξ1 ≥ rdr −∫ 0

−∞Cr nξ1 ≤ rdr = E [nξ1] .


2.9 Variance

Definition 2.20 (Liu and Liu [132]) Let ξ be a fuzzy variable with finiteexpected value e. Then the variance of ξ is defined by V [ξ] = E[(ξ − e)2].

The variance of a fuzzy variable provides a measure of the spread of thedistribution around its expected value.

Example 2.40: Let ξ be an equipossible fuzzy variable (a, b). Then itsexpected value is e = (a+ b)/2, and for any positive number r, we have

Cr(ξ − e)2 ≥ r =

1/2, if r ≤ (b− a)2/4

0, if r > (b− a)2/4.

Thus the variance is

V [ξ] =∫ +∞

0

Cr(ξ − e)2 ≥ rdr =∫ (b−a)2/4

0

12dr =

(b− a)2

8.


Example 2.41: Let ξ = (a, b, c) be a triangular fuzzy variable. Then itsvariance is

V [ξ] =33α3 + 21α2β + 11αβ2 − β3

384α

where α = (b − a) ∨ (c − b) and β = (b − a) ∧ (c − b). Especially, if ξ issymmetric, i.e., b− a = c− b, then its variance is V [ξ] = (c− a)2/24.

Example 2.42: A fuzzy variable ξ is called normally distributed if it has anormal membership function

µ(x) = 2(

1 + exp(π|x− e|√

6σ

))−1

, x ∈ <, σ > 0. (2.70)

The expected value is e and variance is σ2. Let ξ1 and ξ2 be independentlyand normally distributed fuzzy variables with expected values e1 and e2,variances σ2

1 and σ22 , respectively. Then for any real numbers a1 and a2, the

fuzzy variable a1ξ1 + a2ξ2 is also normally distributed with expected valuea1e1 + a2e2 and variance (|a1|σ1 + |a2|σ2)2.

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x

µ(x)

0

1

0.434

ee− σ e+ σ...................................................................

........................................................

...................................

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. . . . . . . . . . . . . . . . . . .

..

..

..

..

..

..

..

..

..

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..........

.........

Figure 2.3: Normal Membership Function

Theorem 2.40 If ξ is a fuzzy variable whose variance exists, a and b arereal numbers, then V [aξ + b] = a2V [ξ].


V [aξ + b] = E[(aξ + b− aE[ξ]− b)2

]= a2E[(ξ − E[ξ])2] = a2V [ξ].

Theorem 2.41 Let ξ be a fuzzy variable with expected value e. Then V [ξ] =0 if and only if Crξ = e = 1.

Proof: If V [ξ] = 0, then E[(ξ − e)2] = 0. Note that

E[(ξ − e)2] =∫ +∞

0

Cr(ξ − e)2 ≥ rdr


which implies Cr(ξ−e)2 ≥ r = 0 for any r > 0. Hence we have Cr(ξ−e)2 =0 = 1, i.e., Crξ = e = 1. Conversely, if Crξ = e = 1, then we haveCr(ξ − e)2 = 0 = 1 and Cr(ξ − e)2 ≥ r = 0 for any r > 0. Thus

V [ξ] =∫ +∞

0

Cr(ξ − e)2 ≥ rdr = 0.


Let ξ be a fuzzy variable that takes values in [a, b], but whose membershipfunction is otherwise arbitrary. When its expected value is given, the maxi-mum variance theorem will provide the maximum variance of ξ, thus playingan important role in treating games against nature.

Theorem 2.42 (Li and Liu [111]) Let f be a convex function on [a, b], andξ a fuzzy variable that takes values in [a, b] and has expected value e. Then

E[f(ξ)] ≤ b− e

b− af(a) +

e− a

b− af(b). (2.71)

Proof: For each θ ∈ Θ, we have a ≤ ξ(θ) ≤ b and

ξ(θ) =b− ξ(θ)b− a

a+ξ(θ)− a

b− ab.


f(ξ(θ)) ≤ b− ξ(θ)b− a

f(a) +ξ(θ)− a

b− af(b).


Theorem 2.43 (Li and Liu [111], Maximum Variance Theorem) Let ξ be afuzzy variable that takes values in [a, b] and has expected value e. Then

V [ξ] ≤ (e− a)(b− e) (2.72)

and equality holds if the fuzzy variable ξ has membership function

µ(x) =

2(b− e)b− a

∧ 1, if x = a

2(e− a)b− a

∧ 1, if x = b.

(2.73)

Proof: It follows from Theorem 2.42 immediately by defining f(x) = (x−e)2.It is also easy to verify that the fuzzy variable determined by (2.73) hasvariance (e− a)(b− e). The theorem is proved.


2.10 Moments

Definition 2.21 (Liu [134]) Let ξ be a fuzzy variable, and k a positive num-ber. Then(a) the expected value E[ξk] is called the kth moment;(b) the expected value E[|ξ|k] is called the kth absolute moment;(c) the expected value E[(ξ − E[ξ])k] is called the kth central moment;(d) the expected value E[|ξ−E[ξ]|k] is called the kth absolute central moment.


Example 2.43: A fuzzy variable ξ is called exponentially distributed if ithas an exponential membership function

µ(x) = 2(

1 + exp(

πx√6m

))−1

, x ≥ 0, m > 0. (2.74)

The expected value is (√

6m ln 2)/π and the second moment is m2. Let ξ1and ξ2 be independently and exponentially distributed fuzzy variables withsecond moments m2

1 and m22, respectively. Then for any positive real numbers

a1 and a2, the fuzzy variable a1ξ1+a2ξ2 is also exponentially distributed withsecond moment (a1m1 + a2m2)2.

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x

µ(x)

0

1

0.434

m

.........................................................................................................................................................................................................................................................................................................................................

. . . . . . . . . ..........

Figure 2.4: Exponential Membership Function

Theorem 2.44 Let ξ be a nonnegative fuzzy variable, and k a positive num-ber. Then the k-th moment

E[ξk] = k

∫ +∞

0

rk−1Crξ ≥ rdr. (2.75)


E[ξk] =∫ ∞

0

Crξk ≥ xdx =∫ ∞

0

Crξ ≥ rdrk = k

∫ ∞

0

rk−1Crξ ≥ rdr.



Theorem 2.45 (Li and Liu [111]) Let ξ be a fuzzy variable that takes val-ues in [a, b] and has expected value e. Then for any positive integer k, thekth absolute moment and kth absolute central moment satisfy the followinginequalities,

E[|ξ|k] ≤ b− e

b− a|a|k +

e− a

b− a|b|k, (2.76)

E[|ξ − e|k] ≤ b− e

b− a(e− a)k +

e− a

b− a(b− e)k. (2.77)


2.11 Critical Values

In order to rank fuzzy variables, we may use two critical values: optimisticvalue and pessimistic value.

Definition 2.22 (Liu [130]) Let ξ be a fuzzy variable, and α ∈ (0, 1]. Then

ξsup(α) = supr∣∣ Cr ξ ≥ r ≥ α

(2.78)

is called the α-optimistic value to ξ, and

ξinf(α) = infr∣∣ Cr ξ ≤ r ≥ α

(2.79)

is called the α-pessimistic value to ξ.

This means that the fuzzy variable ξ will reach upwards of the α-optimisticvalue ξsup(α) with credibility α, and will be below the α-pessimistic valueξinf(α) with credibility α. In other words, the α-optimistic value ξsup(α) isthe supremum value that ξ achieves with credibility α, and the α-pessimisticvalue ξinf(α) is the infimum value that ξ achieves with credibility α.

Example 2.44: Let ξ be an equipossible fuzzy variable on (a, b). Then itsα-optimistic and α-pessimistic values are

ξsup(α) =

b, if α ≤ 0.5a, if α > 0.5,

ξinf(α) =

a, if α ≤ 0.5b, if α > 0.5.

Example 2.45: Let ξ = (a, b, c) be a triangular fuzzy variable. Then itsα-optimistic and α-pessimistic values are

ξsup(α) =

2αb+ (1− 2α)c, if α ≤ 0.5(2α− 1)a+ (2− 2α)b, if α > 0.5,


ξinf(α) =

(1− 2α)a+ 2αb, if α ≤ 0.5(2− 2α)b+ (2α− 1)c, if α > 0.5.

Example 2.46: Let ξ = (a, b, c, d) be a trapezoidal fuzzy variable. Then itsα-optimistic and α-pessimistic values are

ξsup(α) =

2αc+ (1− 2α)d, if α ≤ 0.5(2α− 1)a+ (2− 2α)b, if α > 0.5,

ξinf(α) =

(1− 2α)a+ 2αb, if α ≤ 0.5(2− 2α)c+ (2α− 1)d, if α > 0.5.

Theorem 2.46 Let ξ be a fuzzy variable. If α > 0.5, then we have

Crξ ≤ ξinf(α) ≥ α, Crξ ≥ ξsup(α) ≥ α. (2.80)

Proof: It follows from the definition of α-pessimistic value that there existsa decreasing sequence xi such that Crξ ≤ xi ≥ α and xi ↓ ξinf(α) asi→∞. Since ξ ≤ xi ↓ ξ ≤ ξinf(α) and limi→∞Crξ ≤ xi ≥ α > 0.5, itfollows from the credibility semicontinuity law that

Crξ ≤ ξinf(α) = limi→∞

Crξ ≤ xi ≥ α.

Similarly, there exists an increasing sequence xi such that Crξ ≥ xi ≥δ and xi ↑ ξsup(α) as i → ∞. Since ξ ≥ xi ↓ ξ ≥ ξsup(α) andlimi→∞Crξ ≥ xi ≥ α > 0.5, it follows from the credibility semicontinuitylaw that

Crξ ≥ ξsup(α) = limi→∞

Crξ ≥ xi ≥ α.


Example 2.47: When α ≤ 0.5, it is possible that the inequalities

Crξ ≤ ξinf(α) < α, Crξ ≥ ξsup(α) < α

hold. Let ξ be an equipossible fuzzy variable on (−1, 1). It is clear thatξinf(0.5) = −1. However, Crξ ≤ ξinf(0.5) = 0 < 0.5. In addition,ξsup(0.5) = 1 and Crξ ≥ ξsup(0.5) = 0 < 0.5.

Theorem 2.47 Let ξ be a fuzzy variable. Then we have(a) ξinf(α) is an increasing and left-continuous function of α;(b) ξsup(α) is a decreasing and left-continuous function of α.

Proof: (a) It is easy to prove that ξinf(α) is an increasing function of α.Next, we prove the left-continuity of ξinf(α) with respect to α. Let αi bean arbitrary sequence of positive numbers such that αi ↑ α. Then ξinf(αi)


is an increasing sequence. If the limitation is equal to ξinf(α), then the left-continuity is proved. Otherwise, there exists a number z∗ such that

limi→∞


Thus Crξ ≤ z∗ ≥ αi for each i. Letting i → ∞, we get Crξ ≤ z∗ ≥ α.Hence z∗ ≥ ξinf(α). A contradiction proves the left-continuity of ξinf(α) withrespect to α. The part (b) may be proved similarly.

Theorem 2.48 Let ξ be a fuzzy variable. Then we have(a) if α > 0.5, then ξinf(α) ≥ ξsup(α);(b) if α ≤ 0.5, then ξinf(α) ≤ ξsup(α).


1 ≥ Crξ < ξ(α)+ Crξ > ξ(α) ≥ α+ α > 1.

A contradiction proves ξinf(α) ≥ ξsup(α). Part (b): Assume that ξinf(α) >ξsup(α). It follows from the definition of ξinf(α) that Crξ ≤ ξ(α) < α.Similarly, it follows from the definition of ξsup(α) that Crξ ≥ ξ(α) < α.Thus

1 ≤ Crξ ≤ ξ(α)+ Crξ ≥ ξ(α) < α+ α ≤ 1.

A contradiction proves ξinf(α) ≤ ξsup(α). The theorem is proved.

Theorem 2.49 Let ξ be a fuzzy variable. Then we have(a) if c ≥ 0, then (cξ)sup(α) = cξsup(α) and (cξ)inf(α) = cξinf(α);(b) if c < 0, then (cξ)sup(α) = cξinf(α) and (cξ)inf(α) = cξsup(α).

Proof: If c = 0, then the part (a) is obviously valid. When c > 0, we have

(cξ)sup(α) = sup r | Crcξ ≥ r ≥ α= c sup r/c | Cr ξ ≥ r/c ≥ α= cξsup(α).

A similar way may prove that (cξ)inf(α) = cξinf(α).In order to prove the part (b), it suffices to verify that (−ξ)sup(α) =

−ξinf(α) and (−ξ)inf(α) = −ξsup(α). In fact, for any α ∈ (0, 1], we have

(−ξ)sup(α) = supr | Cr−ξ ≥ r ≥ α= − inf−r | Crξ ≤ −r ≥ α= −ξinf(α).



Theorem 2.50 Suppose that ξ and η are independent fuzzy variables. Thenfor any α ∈ (0, 1], we have

(ξ + η)sup(α) = ξsup(α) + ηsup(α), (ξ + η)inf(α) = ξinf(α) + ηinf(α),

(ξη)sup(α) = ξsup(α)ηsup(α), (ξη)inf(α) = ξinf(α)ηinf(α), if ξ ≥ 0, η ≥ 0,

(ξ ∨ η)sup(α) = ξsup(α) ∨ ηsup(α), (ξ ∨ η)inf(α) = ξinf(α) ∨ ηinf(α),

(ξ ∧ η)sup(α) = ξsup(α) ∧ ηsup(α), (ξ ∧ η)inf(α) = ξinf(α) ∧ ηinf(α).

Proof: For any given number ε > 0, since ξ and η are independent fuzzyvariables, we have

Crξ + η ≥ ξsup(α) + ηsup(α)− ε≥ Cr ξ ≥ ξsup(α)− ε/2 ∩ η ≥ ηsup(α)− ε/2= Crξ ≥ ξsup(α)− ε/2 ∧ Crη ≥ ηsup(α)− ε/2 ≥ α

which implies(ξ + η)sup(α) ≥ ξsup(α) + ηsup(α)− ε. (2.81)

On the other hand, by the independence, we have

Crξ + η ≥ ξsup(α) + ηsup(α) + ε≤ Cr ξ ≥ ξsup(α) + ε/2 ∪ η ≥ ηsup(α) + ε/2= Crξ ≥ ξsup(α) + ε/2 ∨ Crη ≥ ηsup(α) + ε/2 < α

which implies(ξ + η)sup(α) ≤ ξsup(α) + ηsup(α) + ε. (2.82)


ξsup(α) + ηsup(α) + ε ≥ (ξ + η)sup(α) ≥ ξsup(α) + ηsup(α)− ε.

Letting ε → 0, we obtain (ξ + η)sup(α) = ξsup(α) + ηsup(α). The otherequalities may be proved similarly.

Example 2.48: The independence condition cannot be removed in Theo-rem 2.50. For example, let Θ = θ1, θ2, Crθ1 = Crθ2 = 1/2, and letfuzzy variables ξ and η be defined as

ξ(θ) =

0, if θ = θ1

1, if θ = θ2,η(θ) =

1, if θ = θ1

0, if θ = θ2.

However, (ξ + η)sup(0.6) = 1 6= 0 = ξsup(0.6) + ηsup(0.6).


2.12 Entropy

Fuzzy entropy is a measure of uncertainty and has been studied by manyresearchers such as De Luca and Termini [29], Kaufmann [76], Yager [239],Kosko [85], Pal and Pal [188], Bhandari and Pal [7], and Pal and Bezdek[192]. Those definitions of entropy characterize the uncertainty resultingprimarily from the linguistic vagueness rather than resulting from informationdeficiency, and vanishes when the fuzzy variable is an equipossible one.

In order to measure the uncertainty of fuzzy variables, Liu [137] suggestedthat an entropy of fuzzy variables should meet at least the following threebasic requirements:(i) minimum: the entropy of a crisp number is minimum, i.e., 0;(ii) maximum: the entropy of an equipossible fuzzy variable is maximum;(iii) universality: the entropy is applicable not only to finite and infinite casesbut also to discrete and continuous cases.

In order to meet those requirements, Li and Liu [97] provided a new def-inition of fuzzy entropy to characterize the uncertainty resulting from infor-mation deficiency which is caused by the impossibility to predict the specifiedvalue that a fuzzy variable takes.

Entropy of Discrete Fuzzy Variables

Definition 2.23 (Li and Liu [97]) Let ξ be a discrete fuzzy variable takingvalues in x1, x2, · · · . Then its entropy is defined by

H[ξ] =∞∑

i=1

S(Crξ = xi) (2.83)

where S(t) = −t ln t− (1− t) ln(1− t).

Remark 2.6: It is easy to verify that S(t) is a symmetric function aboutt = 0.5, strictly increases on the interval [0, 0.5], strictly decreases on theinterval [0.5, 1], and reaches its unique maximum ln 2 at t = 0.5.

Remark 2.7: It is clear that the entropy depends only on the number ofvalues and their credibilities and does not depend on the actual values thatthe fuzzy variable takes.

Example 2.49: Suppose that ξ is a discrete fuzzy variable taking values inx1, x2, · · · . If there exists some index k such that the membership functionµ(xk) = 1, and 0 otherwise, then its entropy H[ξ] = 0.

Example 2.50: Suppose that ξ is a simple fuzzy variable taking valuesin x1, x2, · · · , xn. If its membership function µ(x) ≡ 1, then its entropyH[ξ] = n ln 2.


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...............

0 0.5 1t

S(t)

ln 2

..................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................

. . . . . . . . . . . . . . . . . . . ...........................

Figure 2.5: Function S(t) = −t ln t− (1− t) ln(1− t)

Theorem 2.51 Suppose that ξ is a discrete fuzzy variable taking values inx1, x2, · · · . Then

H[ξ] ≥ 0 (2.84)

and equality holds if and only if ξ is essentially a crisp number.

Proof: The nonnegativity is clear. In addition, H[ξ] = 0 if and only ifCrξ = xi = 0 or 1 for each i. That is, there exists one and only one indexk such that Crξ = xk = 1, i.e., ξ is essentially a crisp number.

This theorem states that the entropy of a fuzzy variable reaches its min-imum 0 when the fuzzy variable degenerates to a crisp number. In this case,there is no uncertainty.

Theorem 2.52 Suppose that ξ is a simple fuzzy variable taking values inx1, x2, · · · , xn. Then

H[ξ] ≤ n ln 2 (2.85)

and equality holds if and only if ξ is an equipossible fuzzy variable.

Proof: Since the function S(t) reaches its maximum ln 2 at t = 0.5, we have

H[ξ] =n∑

i=1

S(Crξ = xi) ≤ n ln 2

and equality holds if and only if Crξ = xi = 0.5, i.e., µ(xi) ≡ 1 for alli = 1, 2, · · · , n.

This theorem states that the entropy of a fuzzy variable reaches its max-imum when the fuzzy variable is an equipossible one. In this case, there isno preference among all the values that the fuzzy variable will take.


Entropy of Continuous Fuzzy Variables

Definition 2.24 (Li and Liu [97]) Let ξ be a continuous fuzzy variable.Then its entropy is defined by

H[ξ] =∫ +∞

−∞S(Crξ = x)dx (2.86)


For any continuous fuzzy variable ξ with membership function µ, we haveCrξ = x = µ(x)/2 for each x ∈ <. Thus

H[ξ] = −∫ +∞

−∞

(µ(x)

2lnµ(x)

2+(

1− µ(x)2

)ln(

1− µ(x)2

))dx. (2.87)

Example 2.51: Let ξ be an equipossible fuzzy variable (a, b). Then µ(x) = 1if a ≤ x ≤ b, and 0 otherwise. Thus its entropy is

H[ξ] = −∫ b

a

(12

ln12

+(

1− 12

)ln(

1− 12

))dx = (b− a) ln 2.

Example 2.52: Let ξ be a triangular fuzzy variable (a, b, c). Then itsentropy is H[ξ] = (c− a)/2.

Example 2.53: Let ξ be a trapezoidal fuzzy variable (a, b, c, d). Then itsentropy is H[ξ] = (d− a)/2 + (ln 2− 0.5)(c− b).

Example 2.54: Let ξ be an exponentially distributed fuzzy variable withsecond moment m2. Then its entropy is H[ξ] = πm/

√6.

Example 2.55: Let ξ be a normally distributed fuzzy variable with expectedvalue e and variance σ2. Then its entropy is H[ξ] =

√6πσ/3.

Theorem 2.53 Let ξ be a continuous fuzzy variable. Then H[ξ] > 0.

Proof: The positivity is clear. In addition, when a continuous fuzzy variabletends to a crisp number, its entropy tends to the minimum 0. However, acrisp number is not a continuous fuzzy variable.

Theorem 2.54 Let ξ be a continuous fuzzy variable taking values on theinterval [a, b]. Then

H[ξ] ≤ (b− a) ln 2 (2.88)

and equality holds if and only if ξ is an equipossible fuzzy variable (a, b).

Proof: The theorem follows from the fact that the function S(t) reaches itsmaximum ln 2 at t = 0.5.


Theorem 2.55 Let ξ and η be two continuous fuzzy variables with member-ship functions µ(x) and ν(x), respectively. If µ(x) ≤ ν(x) for any x ∈ <,then we have H[ξ] ≤ H[η].

Proof: Since µ(x) ≤ ν(x), we have S(µ(x)/2) ≤ S(ν(x)/2) for any x ∈ <.It follows that H[ξ] ≤ H[η].

Theorem 2.56 Let ξ be a continuous fuzzy variable. Then for any realnumbers a and b, we have H[aξ + b] = |a|H[ξ].

Proof: It follows from the definition of entropy that

H[aξ+b] =∫ +∞

−∞S(Craξ+b = x)dx = |a|

∫ +∞

−∞S(Crξ = y)dy = |a|H[ξ].

Maximum Entropy Principle

Given some constraints, for example, expected value and variance, there areusually multiple compatible membership functions. Which membership func-tion shall we take? The maximum entropy principle attempts to select themembership function that maximizes the value of entropy and satisfies theprescribed constraints.

Theorem 2.25 (Li and Liu [104]) Let ξ be a continuous nonnegative fuzzyvariable with finite second moment m2. Then

H[ξ] ≤ πm√6

(2.89)

and the equality holds if ξ is an exponentially distributed fuzzy variable withsecond moment m2.

Proof: Let µ be the membership function of ξ. Note that µ is a continuousfunction. The proof is based on the following two steps.

Step 1: Suppose that µ is a decreasing function on [0,+∞). For thiscase, we have Crξ ≥ x = µ(x)/2 for any x > 0. Thus the second moment

E[ξ2] =∫ +∞

0

Crξ2 ≥ xdx =∫ +∞

0

2xCrξ ≥ xdx =∫ +∞

0

xµ(x)dx.

The maximum entropy membership function µ should maximize the entropy

−∫ +∞

0

(µ(x)

2lnµ(x)

2+(

1− µ(x)2

)ln(

1− µ(x)2

))dx

subject to the moment constraint∫ +∞

0

xµ(x)dx = m2.


The Lagrangian is

L = −∫ +∞

0

(µ(x)

2lnµ(x)

2+(

1− µ(x)2

)ln(

1− µ(x)2

))dx

−λ(∫ +∞

0

xµ(x)dx−m2

).

The maximum entropy membership function meets Euler-Lagrange equation

12

lnµ(x)

2− 1

2ln(

1− µ(x)2

)+ λx = 0

and has the form µ(x) = 2 (1 + exp(2λx))−1. Substituting it into the momentconstraint, we get

µ∗(x) = 2(

1 + exp(

πx√6m

))−1

, x ≥ 0

which is just the exponential membership function with second moment m2,and the maximum entropy is H[ξ∗] = πm/

√6.

Step 2: Let ξ be a general fuzzy variable with second moment m2. Nowwe define a fuzzy variable ξ via membership function

µ(x) = supy≥x

µ(y), x ≥ 0.

Then µ is a decreasing function on [0,+∞), and

Crξ2 ≥ x =12

supy≥√

x

µ(y) =12

supy≥√

x

supz≥y

µ(z) =12

supz≥√

x

µ(z) ≤ Crξ2 ≥ x

for any x > 0. Thus we have

E[ξ2] =∫ +∞

0

Crξ2 ≥ xdx ≤∫ +∞

0

Crξ2 ≥ xdx = E[ξ2] = m2.

It follows from µ(x) ≤ µ(x) and Step 1 that

H[ξ] ≤ H[ξ] ≤π

√E[ξ2]√

6≤ πm√

6.

The theorem is thus proved.

Theorem 2.26 (Li and Liu [104]) Let ξ be a continuous fuzzy variable withfinite expected value e and variance σ2. Then

H[ξ] ≤√

6πσ3

(2.90)

and the equality holds if ξ is a normally distributed fuzzy variable with expectedvalue e and variance σ2.


Proof: Let µ be the continuous membership function of ξ. The proof isbased on the following two steps.

Step 1: Let µ(x) be a unimodal and symmetric function about x = e.For this case, the variance is

V [ξ] =∫ +∞

0

Cr(ξ − e)2 ≥ xdx =∫ +∞

0

Crξ − e ≥√xdx

=∫ +∞

e

2(x− e)Crξ ≥ xdx =∫ +∞

e

(x− e)µ(x)dx

and the entropy is

H[ξ] = −2∫ +∞

e

(µ(x)

2lnµ(x)

2+(

1− µ(x)2

)ln(

1− µ(x)2

))dx.

The maximum entropy membership function µ should maximize the entropysubject to the variance constraint. The Lagrangian is

L = −2∫ +∞

e

(µ(x)

2lnµ(x)

2+(

1− µ(x)2

)ln(

1− µ(x)2

))dx

−λ(∫ +∞

e

(x− e)µ(x)dx− σ2

).

The maximum entropy membership function meets Euler-Lagrange equation

lnµ(x)

2− ln

(1− µ(x)

2

)+ λ(x− e) = 0

and has the form µ(x) = 2 (1 + exp (λ(x− e)))−1. Substituting it into thevariance constraint, we get

µ∗(x) = 2(

1 + exp(π|x− e|√

6σ

))−1

, x ∈ <

which is just the normal membership function with expected value e andvariance σ2, and the maximum entropy is H[ξ∗] =

√6πσ/3.

Step 2: Let ξ be a general fuzzy variable with expected value e andvariance σ2. We define a fuzzy variable ξ by the membership function

µ(x) =

supy≤x

(µ(y) ∨ µ(2e− y)), if x ≤ e

supy≥x

(µ(y) ∨ µ(2e− y)) , if x > e.


It is easy to verify that µ(x) is a unimodal and symmetric function aboutx = e. Furthermore,

Cr

(ξ − e)2 ≥ r

=12

supx≥e+

√r

µ(x) =12

supx≥e+

√r

supy≥x

(µ(y) ∨ µ(2e− y))

=12

supy≥e+

√r

(µ(y) ∨ µ(2e− y)) =12

sup(y−e)2≥r

µ(y)

≤ Cr(ξ − e)2 ≥ r

for any r > 0. Thus

V [ξ] =∫ +∞

0

Cr(ξ − e)2 ≥ rdr ≤∫ +∞

0

Cr(ξ − e)2 ≥ rdr = σ2.

It follows from µ(x) ≤ µ(x) and Step 1 that

H[ξ] ≤ H[ξ] ≤√

6π√V [ξ]

3≤√

6πσ3

.

The proof is complete.

2.13 Distance

Distance between fuzzy variables has been defined in many ways, for exam-ple, Hausdorff distance (Puri and Ralescu [204], Klement et. al. [81]), andHamming distance (Kacprzyk [73]). However, those definitions have no iden-tification property. In order to overcome this shortage, Liu [135] proposed adefinition of distance as follows.

Definition 2.27 (Liu [135]) The distance between fuzzy variables ξ and η isdefined as

d(ξ, η) = E[|ξ − η|]. (2.91)

Example 2.56: Let ξ and η be equipossible fuzzy variables (a1, b1) and(a2, b2), respectively, and (a1, b1)∩(a2, b2) = ∅. Then |ξ−η| is an equipossiblefuzzy variable on the interval with endpoints |a1 − b2| and |b1 − a2|. Thusthe distance between ξ and η is the expected value of |ξ − η|, i.e.,

d(ξ, η) =12

(|a1 − b2|+ |b1 − a2|) .

Example 2.57: Let ξ = (a1, b1, c1) and η = (a2, b2, c2) be triangular fuzzyvariables such that (a1, c1) ∩ (a2, c2) = ∅. Then

d(ξ, η) =14

(|a1 − c2|+ 2|b1 − b2|+ |c1 − a2|) .


Example 2.58: Let ξ = (a1, b1, c1, d1) and η = (a2, b2, c2, d2) be trapezoidalfuzzy variables such that (a1, d1) ∩ (a2, d2) = ∅. Then

d(ξ, η) =14

(|a1 − d2|+ |b1 − c2|+ |c1 − b2|+ |d1 − a2|) .

Theorem 2.57 (Li and Liu [106]) Let ξ, η, τ be fuzzy variables, and let d(·, ·)be the distance. Then we have(a) (Nonnegativity) d(ξ, η) ≥ 0;(b) (Identification) d(ξ, η) = 0 if and only if ξ = η;(c) (Symmetry) d(ξ, η) = d(η, ξ);(d) (Triangle Inequality) d(ξ, η) ≤ 2d(ξ, τ) + 2d(η, τ).

Proof: The parts (a), (b) and (c) follow immediately from the definition.Now we prove the part (d). It follows from the credibility subadditivitytheorem that

d(ξ, η) =∫ +∞

0

Cr |ξ − η| ≥ rdr

≤∫ +∞

0

Cr |ξ − τ |+ |τ − η| ≥ rdr

≤∫ +∞

0

Cr |ξ − τ | ≥ r/2 ∪ |τ − η| ≥ r/2dr

≤∫ +∞

0

(Cr|ξ − τ | ≥ r/2+ Cr|τ − η| ≥ r/2) dr

=∫ +∞

0

Cr|ξ − τ | ≥ r/2dr +∫ +∞

0

Cr|τ − η| ≥ r/2dr

= 2E[|ξ − τ |] + 2E[|τ − η|] = 2d(ξ, τ) + 2d(τ, η).

Example 2.59: Let Θ = θ1, θ2, θ3 and Crθi = 1/2 for i = 1, 2, 3. Wedefine fuzzy variables ξ, η and τ as follows,

ξ(θ) =

1, if θ 6= θ3

0, otherwise,η(θ) =

−1, if θ 6= θ1

0, otherwise,τ(θ) ≡ 0.

It is easy to verify that d(ξ, τ) = d(τ, η) = 1/2 and d(ξ, η) = 3/2. Thus

d(ξ, η) =32(d(ξ, τ) + d(τ, η)).

2.14 Inequalities

There are several useful inequalities for random variable, such as Markovinequality, Chebyshev inequality, Holder’s inequality, Minkowski inequality,


and Jensen’s inequality. This section introduces the analogous inequalitiesfor fuzzy variable.

Theorem 2.58 (Liu [134]) Let ξ be a fuzzy variable, and f a nonnegativefunction. If f is even and increasing on [0,∞), then for any given numbert > 0, we have

Cr|ξ| ≥ t ≤ E[f(ξ)]f(t)

. (2.92)

Proof: It is clear that Cr|ξ| ≥ f−1(r) is a monotone decreasing functionof r on [0,∞). It follows from the nonnegativity of f(ξ) that

E[f(ξ)] =∫ +∞

0

Crf(ξ) ≥ rdr

=∫ +∞

0

Cr|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

Cr|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

dr · Cr|ξ| ≥ f−1(f(t))

= f(t) · Cr|ξ| ≥ t


Theorem 2.59 (Liu [134], Markov Inequality) Let ξ be a fuzzy variable.Then for any given numbers t > 0 and p > 0, we have

Cr|ξ| ≥ t ≤ E[|ξ|p]tp

. (2.93)


Theorem 2.60 (Liu [134], Chebyshev Inequality) Let ξ be a fuzzy variablewhose variance V [ξ] exists. Then for any given number t > 0, we have

Cr |ξ − E[ξ]| ≥ t ≤ V [ξ]t2

. (2.94)

Proof: It is a special case of Theorem 2.58 when the fuzzy variable ξ isreplaced with ξ − E[ξ], and f(x) = x2.

Theorem 2.61 (Liu [134], Holder’s Inequality) Let p and q be two positivereal numbers with 1/p+1/q = 1, and let ξ and η be independent fuzzy variableswith E[|ξ|p] <∞ and E[|η|q] <∞. Then we have

E[|ξη|] ≤ p√E[|ξ|p] q

√E[|η|q]. (2.95)









E[f(|ξ|p, |η|q)] ≤ f(E[|ξ|p], E[|η|q]).


Theorem 2.62 (Liu [134], Minkowski Inequality) Let p be a real numberwith p ≥ 1, and let ξ and η be independent fuzzy variables with E[|ξ|p] < ∞and E[|η|p] <∞. Then we have

p√E[|ξ + η|p] ≤ p

√E[|ξ|p] + p

√E[|η|p]. (2.96)


√x + p







E[f(|ξ|p, |η|p)] ≤ f(E[|ξ|p], E[|η|p]).


Theorem 2.63 (Liu [135], Jensen’s Inequality) Let ξ be a fuzzy variable,and f : < → < a convex function. If E[ξ] and E[f(ξ)] are finite, then

f(E[ξ]) ≤ E[f(ξ)]. (2.97)




f(ξ)− f(E[ξ]) ≥ k · (ξ − E[ξ]).


E[f(ξ)]− f(E[ξ]) ≥ k · (E[ξ]− E[ξ]) = 0



This section discusses some convergence concepts of fuzzy sequence: conver-gence almost surely (a.s.), convergence in credibility, convergence in mean,and convergence in distribution.

Table 2.1: Relations among Convergence Concepts

Convergence Almost SurelyConvergence

→Convergence

in Mean in Credibility Convergence in Distribution

Definition 2.28 (Liu [134]) Suppose that ξ, ξ1, ξ2, · · · are fuzzy variables de-fined on the credibility space (Θ,P,Cr). The sequence ξi is said to be con-vergent a.s. to ξ if and only if there exists an event A with CrA = 1 suchthat

limi→∞

|ξi(θ)− ξ(θ)| = 0 (2.98)

for every θ ∈ A. In that case we write ξi → ξ, a.s.

Definition 2.29 (Liu [134]) Suppose that ξ, ξ1, ξ2, · · · are fuzzy variables de-fined on the credibility space (Θ,P,Cr). We say that the sequence ξi con-verges in credibility to ξ if

limi→∞

Cr |ξi − ξ| ≥ ε = 0 (2.99)

for every ε > 0.

Definition 2.30 (Liu [134]) Suppose that ξ, ξ1, ξ2, · · · are fuzzy variableswith finite expected values defined on the credibility space (Θ,P,Cr). Wesay that the sequence ξi converges in mean to ξ if

limi→∞

E[|ξi − ξ|] = 0. (2.100)



limi→∞

E[|ξi − ξ|2] = 0. (2.101)

Definition 2.31 (Liu [134]) Suppose that Φ,Φ1,Φ2, · · · are the credibilitydistributions of fuzzy variables ξ, ξ1, ξ2, · · · , respectively. We say that ξiconverges in distribution to ξ if Φi → Φ at any continuity point of Φ.

Convergence in Mean vs. Convergence in Credibility

Theorem 2.64 (Liu [134]) Suppose that ξ, ξ1, ξ2, · · · are fuzzy variables de-fined on the credibility space (Θ,P,Cr). If the sequence ξi converges inmean to ξ, then ξi converges in credibility to ξ.

Proof: It follows from Theorem 2.59 that, for any given number ε > 0,

Cr |ξi − ξ| ≥ ε ≤ E[|ξi − ξ|]ε

→ 0

as i→∞. Thus ξi converges in credibility to ξ.

Example 2.60: Convergence in credibility does not imply convergence inmean. For example, take (Θ,P,Cr) to be θ1, θ2, · · · with Crθ1 = 1/2 andCrθj = 1/j for j = 2, 3, · · · The fuzzy variables are defined by

ξi(θj) =

i, if j = i

0, otherwise


Cr |ξi − ξ| ≥ ε =1i→ 0.

That is, the sequence ξi converges in credibility to ξ. However,

E [|ξi − ξ|] ≡ 1 6→ 0.


Convergence Almost Surely vs. Convergence in Credibility

Example 2.61: Convergence a.s. does not imply convergence in credibility.For example, take (Θ,P,Cr) to be θ1, θ2, · · · with Crθj = j/(2j + 1) forj = 1, 2, · · · The fuzzy variables are defined by

ξi(θj) =

i, if j = i

0, otherwise


for i = 1, 2, · · · and ξ = 0. Then the sequence ξi converges a.s. to ξ.However, for any small number ε > 0, we have

Cr |ξi − ξ| ≥ ε =i

2i+ 1→ 1

2.

That is, the sequence ξi does not converge in credibility to ξ.

Theorem 2.65 (Wang and Liu [233]) Suppose that ξ, ξ1, ξ2, · · · are fuzzyvariables defined on the credibility space (Θ,P,Cr). If the sequence ξi con-verges in credibility to ξ, then ξi converges a.s. to ξ.

Proof: If ξi does not converge a.s. to ξ, then there exists an elementθ∗ ∈ Θ with Crθ∗ > 0 such that ξi(θ∗) 6→ ξ(θ∗) as i→∞. In other words,there exists a small number ε > 0 and a subsequence ξik

(θ∗) such that|ξik

(θ∗)− ξ(θ∗)| ≥ ε for any k. Since credibility measure is an increasing setfunction, we have

Cr |ξik− ξ| ≥ ε ≥ Crθ∗ > 0

for any k. It follows that ξi does not converge in credibility to ξ. Acontradiction proves the theorem.

Convergence in Credibility vs. Convergence in Distribution

Theorem 2.66 (Wang and Liu [233]) Suppose that ξ, ξ1, ξ2, · · · are fuzzyvariables. If the sequence ξi converges in credibility to ξ, then ξi con-verges in distribution to ξ.

Proof: Let x be any given continuity point of the distribution Φ. On theone hand, for any y > x, we have

ξi ≤ x = ξi ≤ x, ξ ≤ y ∪ ξi ≤ x, ξ > y ⊂ ξ ≤ y ∪ |ξi − ξ| ≥ y − x.

It follows from the credibility subadditivity theorem that

Φi(x) ≤ Φ(y) + Cr|ξi − ξ| ≥ y − x.

Since ξi converges in credibility to ξ, we have Cr|ξi − ξ| ≥ y − x → 0.Thus we obtain lim supi→∞Φi(x) ≤ Φ(y) for any y > x. Letting y → x, weget

lim supi→∞

Φi(x) ≤ Φ(x). (2.102)


ξ ≤ z = ξ ≤ z, ξi ≤ x ∪ ξ ≤ z, ξi > x ⊂ ξi ≤ x ∪ |ξi − ξ| ≥ x− z

which implies that

Φ(z) ≤ Φi(x) + Cr|ξi − ξ| ≥ x− z.


Since Cr|ξi − ξ| ≥ x − z → 0, we obtain Φ(z) ≤ lim infi→∞Φi(x) for anyz < x. Letting z → x, we get


Φi(x). (2.103)

It follows from (2.102) and (2.103) that Φi(x) → Φ(x). The theorem isproved.

Example 2.62: Convergence in distribution does not imply convergencein credibility. For example, take (Θ,P,Cr) to be θ1, θ2 with Crθ1 =Crθ2 = 1/2, and define

ξ(θ) =−1, if θ = θ11, if θ = θ2.

We also define ξi = −ξ for i = 1, 2, · · · Then ξi and ξ are identically dis-tributed. Thus ξi converges in distribution to ξ. But, for any small numberε > 0, we have Cr|ξi − ξ| > ε = CrΘ = 1. That is, the sequence ξidoes not converge in credibility to ξ.

Convergence Almost Surely vs. Convergence in Distribution

Example 2.63: Convergence in distribution does not imply convergence a.s.For example, take (Θ,P,Cr) to be θ1, θ2 with Crθ1 = Crθ2 = 1/2, anddefine

ξ(θ) =−1, if θ = θ11, if θ = θ2.

We also define ξi = −ξ for i = 1, 2, · · · Then ξi converges in distributionto ξ. However, ξi does not converge a.s. to ξ.

Example 2.64: Convergence a.s. does not imply convergence in distribution.For example, take (Θ,P,Cr) to be θ1, θ2, · · · with Crθj = j/(2j + 1) forj = 1, 2, · · · The fuzzy variables are defined by

ξi(θj) =

i, if j = i

0, otherwise

for i = 1, 2, · · · and ξ = 0. Then the sequence ξi converges a.s. to ξ.However, the credibility distributions of ξi are

Φi(x) =

0, if x < 0

(i+ 1)/(2i+ 1), if 0 ≤ x < i

1, if x ≥ i,

i = 1, 2, · · · , respectively. The credibility distribution of ξ is

Φ(x) =

0, if x < 01, if x ≥ 0.


It is clear that Φi(x) 6→ Φ(x) at x > 0. That is, the sequence ξi does notconverge in distribution to ξ.

2.16 Conditional Credibility

We now consider the credibility of an event A after it has been learned thatsome other event B has occurred. This new credibility of A is called theconditional credibility of A given B.

In order to define a conditional credibility measure CrA|B, at first wehave to enlarge CrA ∩ B because CrA ∩ B < 1 for all events wheneverCrB < 1. It seems that we have no alternative but to divide CrA∩B byCrB. Unfortunately, CrA∩B/CrB is not always a credibility measure.However, the value CrA|B should not be greater than CrA ∩ B/CrB(otherwise the normality will be lost), i.e.,

CrA|B ≤ CrA ∩BCrB

. (2.104)

On the other hand, in order to preserve the self-duality, we should have

CrA|B = 1− CrAc|B ≥ 1− CrAc ∩BCrB

. (2.105)

Furthermore, since (A ∩B) ∪ (Ac ∩B) = B, we have CrB ≤ CrA ∩B+CrAc ∩B by using the credibility subadditivity theorem. Thus

0 ≤ 1− CrAc ∩BCrB

≤ CrA ∩BCrB

≤ 1. (2.106)

Hence any numbers between 1−CrAc ∩B/CrB and CrA∩B/CrBare reasonable values that the conditional credibility may take. Based onthe maximum uncertainty principle (see Appendix E), we have the followingconditional credibility measure.

Definition 2.32 (Liu [138]) Let (Θ,P,Cr) be a credibility space, and A,B ∈P. Then the conditional credibility measure of A given B is defined by

CrA|B =

CrA ∩BCrB

, ifCrA ∩B

CrB< 0.5

1− CrAc ∩BCrB

, ifCrAc ∩B

CrB< 0.5

0.5, otherwise

(2.107)

provided that CrB > 0.

Section 2.16 - Conditional Credibility 109

It follows immediately from the definition of conditional credibility that

1− CrAc ∩BCrB

≤ CrA|B ≤ CrA ∩BCrB

. (2.108)

Furthermore, the value of CrA|B takes values as close to 0.5 as possiblein the interval. In other words, it accords with the maximum uncertaintyprinciple.

Theorem 2.67 Let (Θ,P,Cr) be a credibility space, and B an event withCrB > 0. Then Cr·|B defined by (2.107) is a credibility measure, and(Θ,P,Cr·|B) is a credibility space.

Proof: It is sufficient to prove that Cr·|B satisfies the normality, mono-tonicity, self-duality and maximality axioms. At first, it satisfies the normal-ity axiom, i.e.,

CrΘ|B = 1− CrΘc ∩BCrB

= 1− Cr∅CrB

= 1.

For any events A1 and A2 with A1 ⊂ A2, if

CrA1 ∩BCrB

≤ CrA2 ∩BCrB

< 0.5,

then

CrA1|B =CrA1 ∩B

CrB≤ CrA2 ∩B

CrB= CrA2|B.

IfCrA1 ∩B

CrB≤ 0.5 ≤ CrA2 ∩B

CrB,

then CrA1|B ≤ 0.5 ≤ CrA2|B. If

0.5 <CrA1 ∩B

CrB≤ CrA2 ∩B

CrB,

then we have

CrA1|B =(1− CrAc

1 ∩BCrB

)∨0.5 ≤

(1− CrAc

2 ∩BCrB

)∨0.5 = CrA2|B.

This means that Cr·|B satisfies the monotonicity axiom. For any event A,if

CrA ∩BCrB

≥ 0.5,CrAc ∩B

CrB≥ 0.5,

then we have CrA|B+ CrAc|B = 0.5 + 0.5 = 1 immediately. Otherwise,without loss of generality, suppose

CrA ∩BCrB

< 0.5 <CrAc ∩B

CrB,


then we have

CrA|B+ CrAc|B =CrA ∩B

CrB+(

1− CrA ∩BCrB

)= 1.

That is, Cr·|B satisfies the self-duality axiom. Finally, for any events Aiwith supi CrAi|B < 0.5, we have supi CrAi ∩B < 0.5 and

supi

CrAi|B =supi CrAi ∩B

CrB=

Cr∪iAi ∩BCrB

= Cr∪iAi|B.

Thus Cr·|B satisfies the maximality axiom. Hence Cr·|B is a credibilitymeasure. Furthermore, (Θ,P,Cr·|B) is a credibility space.

Example 2.65: Let ξ be a fuzzy variable, and X a set of real numbers suchthat Crξ ∈ X > 0. Then for any x ∈ X, the conditional credibility ofξ = x given ξ ∈ X is

Cr ξ = x|ξ ∈ X =

Crξ = xCrξ ∈ X

, ifCrξ = xCrξ ∈ X

< 0.5

1− Crξ 6= x, ξ ∈ XCrξ ∈ X

, ifCrξ 6= x, ξ ∈ X

Crξ ∈ X< 0.5

0.5, otherwise.

Example 2.66: Let ξ and η be two fuzzy variables, and Y a set of realnumbers such that Crη ∈ Y > 0. Then we have

Cr ξ = x|η ∈ Y =

Crξ = x, η ∈ Y Crη ∈ Y

, ifCrξ = x, η ∈ Y

Crη ∈ Y < 0.5

1− Crξ 6= x, η ∈ Y Crη ∈ Y

, ifCrξ 6= x, η ∈ Y

Crη ∈ Y < 0.5

0.5, otherwise.

Definition 2.33 (Liu [138]) The conditional membership function of a fuzzyvariable ξ given B is defined by

µ(x|B) = (2Crξ = x|B) ∧ 1, x ∈ < (2.109)


Example 2.67: Let ξ be a fuzzy variable with membership function µ(x),and X a set of real numbers such that µ(x) > 0 for some x ∈ X. Then theconditional membership function of ξ given ξ ∈ X is

µ(x|X) =

2µ(x)supx∈X

µ(x)∧ 1, if sup

x∈Xµ(x) < 1

2µ(x)2− sup

x∈Xc

µ(x)∧ 1, if sup

x∈Xµ(x) = 1

(2.110)


for x ∈ X. Please mention that µ(x|X) ≡ 0 if x 6∈ X.

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x

µ(x|X)

0

1

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0

1

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µ(x|X)

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Figure 2.6: Conditional Membership Function µ(x|X)

Example 2.68: Let ξ and η be two fuzzy variables with joint member-ship function µ(x, y), and Y a set of real numbers. Then the conditionalmembership function of ξ given η ∈ Y is

µ(x|Y ) =

2 supy∈Y

µ(x, y)

supx∈<,y∈Y

µ(x, y)∧ 1, if sup

x∈<,y∈Yµ(x, y) < 1

2 supy∈Y

µ(x, y)

2− supx∈<,y∈Y c


x∈<,y∈Yµ(x, y) = 1

(2.111)

provided that µ(x, y) > 0 for some x ∈ < and y ∈ Y . Especially, theconditional membership function of ξ given η = y is

µ(x|y) =

2µ(x, y)supx∈<


x∈<µ(x, y) < 1

2µ(x, y)2− sup

x∈<,z 6=yµ(x, z)

∧ 1, if supx∈<

µ(x, y) = 1

provided that µ(x, y) > 0 for some x ∈ <.

Definition 2.34 Let ξ be a fuzzy variable on (Θ,P,Cr). A conditional fuzzyvariable of ξ given B is a function ξ|B from the conditional credibility space(Θ,P,Cr·|B) to the set of real numbers such that

ξ|B(θ) ≡ ξ(θ), ∀θ ∈ Θ. (2.112)


Definition 2.35 (Liu [138]) The conditional credibility distribution Φ: < →[0, 1] of a fuzzy variable ξ given B is defined by

Φ(x|B) = Cr ξ ≤ x|B (2.113)


Example 2.69: Let ξ and η be fuzzy variables. Then the conditional credi-bility distribution of ξ given η = y is

Φ(x|η = y) =

Crξ ≤ x, η = yCrη = y

, ifCrξ ≤ x, η = y

Crη = y< 0.5

1− Crξ > x, η = yCrη = y

, ifCrξ > x, η = y

Crη = y< 0.5

0.5, otherwise

provided that Crη = y > 0.

Definition 2.36 (Liu [138]) The conditional credibility density function φof a fuzzy variable ξ given B is a nonnegative function such that

Φ(x|B) =∫ x

−∞φ(y|B)dy, ∀x ∈ <, (2.114)

∫ +∞

−∞φ(y|B)dy = 1 (2.115)

where Φ(x|B) is the conditional credibility distribution of ξ given B.

Definition 2.37 (Liu [138]) Let ξ be a fuzzy variable. Then the conditionalexpected value of ξ given B is defined by

E[ξ|B] =∫ +∞

0

Crξ ≥ r|Bdr −∫ 0

−∞Crξ ≤ r|Bdr (2.116)


Following conditional credibility and conditional expected value, we alsohave conditional variance, conditional moments, conditional critical values,conditional entropy as well as conditional convergence.

Hazard Rate

Definition 2.38 Let ξ be a nonnegative fuzzy variable representing lifetime.Then the hazard rate (or failure rate) is

h(x) = lim∆↓0

Crξ ≤ x+ ∆∣∣ ξ > x. (2.117)


The hazard rate tells us the credibility of a failure just after time x when itis functioning at time x.

Example 2.70: Let ξ be an exponentially distributed fuzzy variable. Thenits hazard rate h(x) ≡ 0.5. In fact, the hazard rate is always 0.5 if themembership function is positive and decreasing.

Example 2.71: Let ξ be a triangular fuzzy variable (a, b, c) with a ≥ 0.Then its hazard rate

h(x) =

0, if x ≤ a

x− a

b− a, if a ≤ x ≤ (a+ b)/2

0.5, if (a+ b)/2 ≤ x < c

0, if x ≥ c.

Chapter 3

Chance Theory

Fuzziness and randomness are two basic types of uncertainty. In many cases,fuzziness and randomness simultaneously appear in a system. In order todescribe this phenomena, a fuzzy random variable was introduced by Kwak-ernaak [87] as a random element taking “fuzzy variable” values. In addition,a random fuzzy variable was proposed by Liu [130] as a fuzzy element taking“random variable” values. For example, it might be known that the lifetimeof a modern engine is an exponentially distributed random variable with anunknown parameter. If the parameter is provided as a fuzzy variable, thenthe lifetime is a random fuzzy variable.

More generally, a hybrid variable was introduced by Liu [136] in 2006 as atool to describe the quantities with fuzziness and randomness. Fuzzy randomvariable and random fuzzy variable are instances of hybrid variable. In orderto measure hybrid events, a concept of chance measure was introduced by Liand Liu [107] in 2009. Chance theory is a hybrid of probability theory andcredibility theory. Perhaps the reader would like to know what axioms weshould assume for chance theory. In fact, chance theory will be based on thethree axioms of probability and five axioms of credibility.

The emphasis in this chapter is mainly on chance space, hybrid variable,chance measure, chance distribution, expected value, variance, moments,critical values, entropy, distance, convergence almost surely, convergence inchance, convergence in mean, convergence in distribution, and conditionalchance.

3.1 Chance Space

Chance theory begins with the concept of chance space that inherits themathematical foundations of both probability theory and credibility theory.

Definition 3.1 (Liu [136]) Suppose that (Θ,P,Cr) is a credibility space and(Ω,A,Pr) is a probability space. The product (Θ,P,Cr)× (Ω,A,Pr) is called

116 Chapter 3 - Chance Theory

a chance space.

The universal set Θ×Ω is clearly the set of all ordered pairs of the form(θ, ω), where θ ∈ Θ and ω ∈ Ω. What is the product σ-algebra P×A? Whatis the product measure Cr× Pr? Let us discuss these two basic problems.

What is the product σ-algebra P×A?

Generally speaking, it is not true that all subsets of Θ × Ω are measurable.Let Λ be a subset of Θ× Ω. Write

Λ(θ) =ω ∈ Ω

∣∣ (θ, ω) ∈ Λ. (3.1)

It is clear that Λ(θ) is a subset of Ω. If Λ(θ) ∈ A holds for each θ ∈ Θ, thenΛ may be regarded as a measurable set.

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Λ

Θ

Ω

θ

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Λ(θ).......................................................................................

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Figure 3.1: Θ× Ω, Λ and Λ(θ)

Definition 3.2 (Liu [138]) Let (Θ,P,Cr)× (Ω,A,Pr) be a chance space. Asubset Λ ⊂ Θ× Ω is called an event if Λ(θ) ∈ A for each θ ∈ Θ.

Example 3.1: Empty set ∅ and universal set Θ× Ω are clearly events.

Example 3.2: Let X ∈ P and Y ∈ A. Then X × Y is a subset of Θ × Ω.Since the set

(X × Y )(θ) =

Y, if θ ∈ X∅, if θ ∈ Xc

is in the σ-algebra A for each θ ∈ Θ, the rectangle X × Y is an event.

Theorem 3.1 (Liu [138]) Let (Θ,P,Cr)× (Ω,A,Pr) be a chance space. Theclass of all events is a σ-algebra over Θ× Ω, and denoted by P×A.

Section 3.1 - Chance Space 117

Proof: At first, it is obvious that Θ×Ω ∈ P×A. For any event Λ, we alwayshave

Λ(θ) ∈ A, ∀θ ∈ Θ.

Thus for each θ ∈ Θ, the set

Λc(θ) =ω ∈ Ω

∣∣ (θ, ω) ∈ Λc

= (Λ(θ))c ∈ A

which implies that Λc ∈ P × A. Finally, let Λ1,Λ2, · · · be events. Then foreach θ ∈ Θ, we have( ∞⋃

i=1

Λi

)(θ) =

ω ∈ Ω

∣∣ (θ, ω) ∈∞⋃

i=1

Λi

=

∞⋃i=1

ω ∈ Ω

∣∣ (θ, ω) ∈ Λi

∈ A.

That is, the countable union ∪iΛi ∈ P×A. Hence P×A is a σ-algebra.

Example 3.3: When Θ× Ω is countable, usually P×A is the power set.

Example 3.4: When Θ×Ω is uncountable, for example Θ×Ω = <2, usuallyP×A is a σ-algebra between the Borel algebra and power set of <2. Let Xbe a nonempty Borel set and let Y be a non-Borel set of real numbers. Itfollows from X × Y 6∈ P×A that P×A is smaller than the power set. It isalso clear that Y ×X ∈ P×A but Y ×X is not a Borel set. Hence P×A islarger than the Borel algebra.

What is the product measure Cr× Pr?

Product probability is a probability measure, and product credibility is acredibility measure. What is the product measure Cr × Pr? We will call itchance measure and define it as follows.

Definition 3.3 (Li and Liu [107]) Let (Θ,P,Cr) × (Ω,A,Pr) be a chancespace. Then a chance measure of an event Λ is defined as

ChΛ =

supθ∈Θ

(Crθ ∧ PrΛ(θ)),

if supθ∈Θ

(Crθ ∧ PrΛ(θ)) < 0.5

1− supθ∈Θ

(Crθ ∧ PrΛc(θ)),

if supθ∈Θ

(Crθ ∧ PrΛ(θ)) ≥ 0.5.

(3.2)

Example 3.5: Take a credibility space (Θ,P,Cr) to be θ1, θ2 with Crθ1 =0.6 and Crθ2 = 0.4, and take a probability space (Ω,A,Pr) to be ω1, ω2with Prω1 = 0.7 and Prω2 = 0.3. Then

Ch(θ1, ω1) = 0.6, Ch(θ2, ω2) = 0.3.


Theorem 3.2 Let (Θ,P,Cr)×(Ω,A,Pr) be a chance space and Ch a chancemeasure. Then we have

Ch∅ = 0, (3.3)

ChΘ× Ω = 1, (3.4)

0 ≤ ChΛ ≤ 1 (3.5)

for any event Λ.

Proof: It follows from the definition immediately.

Theorem 3.3 Let (Θ,P,Cr)×(Ω,A,Pr) be a chance space and Ch a chancemeasure. Then for any event Λ, we have

supθ∈Θ

(Crθ ∧ PrΛ(θ)) ∨ supθ∈Θ

(Crθ ∧ PrΛc(θ)) ≥ 0.5, (3.6)

supθ∈Θ

(Crθ ∧ PrΛ(θ)) + supθ∈Θ

(Crθ ∧ PrΛc(θ)) ≤ 1, (3.7)

supθ∈Θ

(Crθ ∧ PrΛ(θ)) ≤ ChΛ ≤ 1− supθ∈Θ

(Crθ ∧ PrΛc(θ)). (3.8)

Proof: It follows from the basic properties of probability and credibility that

supθ∈Θ

(Crθ ∧ PrΛ(θ)) ∨ supθ∈Θ

(Crθ ∧ PrΛc(θ))

≥ supθ∈Θ

(Crθ ∧ (PrΛ(θ) ∨ PrΛc(θ)))

≥ supθ∈Θ

Crθ ∧ 0.5 = 0.5

andsupθ∈Θ

(Crθ ∧ PrΛ(θ)) + supθ∈Θ

(Crθ ∧ PrΛc(θ))

= supθ1,θ2∈Θ

(Crθ1 ∧ PrΛ(θ1)+ Crθ2 ∧ PrΛc(θ2))

≤ supθ1 6=θ2

(Crθ1+ Crθ2) ∨ supθ∈Θ

(PrΛ(θ)+ PrΛc(θ))

≤ 1 ∨ 1 = 1.

The inequalities (3.8) follows immediately from the above inequalities andthe definition of chance measure.

Theorem 3.4 (Li and Liu [107]) The chance measure is increasing. Thatis,

ChΛ1 ≤ ChΛ2 (3.9)

for any events Λ1 and Λ2 with Λ1 ⊂ Λ2.


Proof: Since Λ1(θ) ⊂ Λ2(θ) and Λc2(θ) ⊂ Λc

1(θ) for each θ ∈ Θ, we have

supθ∈Θ

(Crθ ∧ PrΛ1(θ)) ≤ supθ∈Θ

(Crθ ∧ PrΛ2(θ)),

supθ∈Θ

(Crθ ∧ PrΛc2(θ)) ≤ sup

θ∈Θ(Crθ ∧ PrΛc

1(θ)).

The argument breaks down into three cases.Case 1: sup

θ∈Θ(Crθ ∧ PrΛ2(θ)) < 0.5. For this case, we have

supθ∈Θ

(Crθ ∧ PrΛ1(θ)) < 0.5,

ChΛ2 = supθ∈Θ

(Crθ ∧ PrΛ2(θ)) ≥ supθ∈Θ

(Crθ ∧ PrΛ1(θ) = ChΛ1.

Case 2: supθ∈Θ

(Crθ∧PrΛ2(θ)) ≥ 0.5 and supθ∈Θ

(Crθ∧PrΛ1(θ)) < 0.5.

It follows from Theorem 3.3 that

ChΛ2 ≥ supθ∈Θ

(Crθ ∧ PrΛ2(θ)) ≥ 0.5 > ChΛ1.

Case 3: supθ∈Θ

(Crθ∧PrΛ2(θ)) ≥ 0.5 and supθ∈Θ

(Crθ∧PrΛ1(θ)) ≥ 0.5.

For this case, we have

ChΛ2 = 1−supθ∈Θ

(Crθ∧PrΛc2(θ)) ≥ 1−sup

θ∈Θ(Crθ∧PrΛc

1(θ)) = ChΛ1.

Thus Ch is an increasing measure.

Theorem 3.5 (Li and Liu [107]) The chance measure is self-dual. That is,

ChΛ+ ChΛc = 1 (3.10)

for any event Λ.

Proof: For any event Λ, please note that

ChΛc =

supθ∈Θ

(Crθ ∧ PrΛc(θ)), if supθ∈Θ

(Crθ ∧ PrΛc(θ)) < 0.5

1− supθ∈Θ

(Crθ ∧ PrΛ(θ)), if supθ∈Θ

(Crθ ∧ PrΛc(θ)) ≥ 0.5.

The argument breaks down into three cases.Case 1: sup

θ∈Θ(Crθ ∧ PrΛ(θ)) < 0.5. For this case, we have

supθ∈Θ

(Crθ ∧ PrΛc(θ)) ≥ 0.5,

ChΛ+ChΛc = supθ∈Θ

(Crθ∧PrΛ(θ))+1− supθ∈Θ

(Crθ∧PrΛ(θ)) = 1.


Case 2: supθ∈Θ

(Crθ ∧PrΛ(θ)) ≥ 0.5 and supθ∈Θ

(Crθ ∧PrΛc(θ)) < 0.5.

For this case, we have

ChΛ+ChΛc = 1−supθ∈Θ

(Crθ∧PrΛc(θ))+supθ∈Θ

(Crθ∧PrΛc(θ)) = 1.

Case 3: supθ∈Θ

(Crθ ∧PrΛ(θ)) ≥ 0.5 and supθ∈Θ

(Crθ ∧PrΛc(θ)) ≥ 0.5.

For this case, it follows from Theorem 3.3 that

supθ∈Θ

(Crθ ∧ PrΛ(θ)) = supθ∈Θ

(Crθ ∧ PrΛc(θ)) = 0.5.

Hence ChΛ+ ChΛc = 0.5 + 0.5 = 1. The theorem is proved.

Theorem 3.6 (Li and Liu [107]) For any event X × Y , we have

ChX × Y = CrX ∧ PrY . (3.11)

Proof: The argument breaks down into three cases.Case 1: CrX < 0.5. For this case, we have

supθ∈X

Crθ ∧ PrY = CrX ∧ CrY < 0.5,

ChX × Y = supθ∈X

Crθ ∧ PrY = CrX ∧ PrY .

Case 2: CrX ≥ 0.5 and PrY < 0.5. Then we have

supθ∈X

Crθ ≥ 0.5,

supθ∈X

Crθ ∧ PrY = PrY < 0.5,

ChX × Y = supθ∈X

Crθ ∧ PrY = PrY = CrX ∧ PrY .

Case 3: CrX ≥ 0.5 and PrY ≥ 0.5. Then we have

supθ∈Θ

(Crθ ∧ Pr(X × Y )(θ)) ≥ supθ∈X

Crθ ∧ PrY ≥ 0.5,

ChX × Y = 1− supθ∈Θ

(Crθ ∧ Pr(X × Y )c(θ)) = CrX ∧ PrY .


Example 3.6: It follows from Theorem 3.6 that for any events X × Ω andΘ× Y , we have

ChX × Ω = CrX, ChΘ× Y = PrY . (3.12)


Theorem 3.7 (Li and Liu [107], Chance Subadditivity Theorem) The chancemeasure is subadditive. That is,

ChΛ1 ∪ Λ2 ≤ ChΛ1+ ChΛ2 (3.13)

for any events Λ1 and Λ2. In fact, chance measure is not only finitely sub-additive but also countably subadditive.

Proof: The proof breaks down into three cases.Case 1: ChΛ1 ∪ Λ2 < 0.5. Then ChΛ1 < 0.5, ChΛ2 < 0.5 and

ChΛ1 ∪ Λ2 = supθ∈Θ

(Crθ ∧ Pr(Λ1 ∪ Λ2)(θ))

≤ supθ∈Θ

(Crθ ∧ (PrΛ1(θ)+ PrΛ2(θ)))

≤ supθ∈Θ

(Crθ ∧ PrΛ1(θ)+ Crθ ∧ PrΛ2(θ))

≤ supθ∈Θ

(Crθ ∧ PrΛ1(θ)) + supθ∈Θ

(Crθ ∧ PrΛ2(θ))

= ChΛ1+ ChΛ2.

Case 2: ChΛ1 ∪ Λ2 ≥ 0.5 and ChΛ1 ∨ ChΛ2 < 0.5. We first have

supθ∈Θ

(Crθ ∧ Pr(Λ1 ∪ Λ2)(θ)) ≥ 0.5.

For any sufficiently small number ε > 0, there exists a point θ such that

Crθ ∧ Pr(Λ1 ∪ Λ2)(θ) > 0.5− ε > ChΛ1 ∨ ChΛ2,

Crθ > 0.5− ε > PrΛ1(θ),Crθ > 0.5− ε > PrΛ2(θ).

Thus we have

Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ)+ Crθ ∧ PrΛ1(θ)+ Crθ ∧ PrΛ2(θ)

= Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ)+ PrΛ1(θ)+ PrΛ2(θ)

≥ Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ)+ Pr(Λ1 ∪ Λ2)(θ) ≥ 1− 2ε

because if Crθ ≥ Pr(Λ1 ∪ Λ2)c(θ), then

Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ)+ Pr(Λ1 ∪ Λ2)(θ)

= Pr(Λ1 ∪ Λ2)c(θ)+ Pr(Λ1 ∪ Λ2)(θ)

= 1 ≥ 1− 2ε

and if Crθ < Pr(Λ1 ∪ Λ2)c(θ), then

Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ)+ Pr(Λ1 ∪ Λ2)(θ)

= Crθ+ Pr(Λ1 ∪ Λ2)(θ)

≥ (0.5− ε) + (0.5− ε) = 1− 2ε.


Taking supremum on both sides and letting ε→ 0, we obtain

ChΛ1 ∪ Λ2 = 1− supθ∈Θ

(Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ))

≤ supθ∈Θ

(Crθ ∧ PrΛ1(θ)) + supθ∈Θ

(Crθ ∧ PrΛ2(θ))

= ChΛ1+ ChΛ2.

Case 3: ChΛ1 ∪ Λ2 ≥ 0.5 and ChΛ1 ∨ ChΛ2 ≥ 0.5. Without lossof generality, suppose ChΛ1 ≥ 0.5. For each θ, we first have

Crθ ∧ PrΛc1(θ) = Crθ ∧ Pr(Λc

1(θ) ∩ Λc2(θ)) ∪ (Λc

1(θ) ∩ Λ2(θ))

≤ Crθ ∧ (Pr(Λ1 ∪ Λ2)c(θ)+ PrΛ2(θ))

≤ Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ)+ Crθ ∧ PrΛ2(θ),

i.e., Crθ∧Pr(Λ1 ∪Λ2)c(θ) ≥ Crθ∧PrΛc1(θ)−Crθ∧PrΛ2(θ). It

follows from Theorem 3.3 that

ChΛ1 ∪ Λ2 = 1− supθ∈Θ

(Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ))

≤ 1− supθ∈Θ

(Crθ ∧ PrΛc1(θ)) + sup

θ∈Θ(Crθ ∧ PrΛ2(θ))

≤ ChΛ1+ ChΛ2.


Remark 3.1: For any events Λ1 and Λ2, it follows from the chance subaddi-tivity theorem that the chance measure is null-additive, i.e., ChΛ1 ∪ Λ2 =ChΛ1+ ChΛ2 if either ChΛ1 = 0 or ChΛ2 = 0.

Theorem 3.8 Let Λi be a decreasing sequence of events with ChΛi → 0as i→∞. Then for any event Λ, we have

limi→∞

ChΛ ∪ Λi = limi→∞

ChΛ\Λi = ChΛ. (3.14)

Proof: Since chance measure is increasing and subadditive, we immediatelyhave

ChΛ ≤ ChΛ ∪ Λi ≤ ChΛ+ ChΛi

for each i. Thus we get ChΛ ∪ Λi → ChΛ by using ChΛi → 0. Since(Λ\Λi) ⊂ Λ ⊂ ((Λ\Λi) ∪ Λi), we have

ChΛ\Λi ≤ ChΛ ≤ ChΛ\Λi+ ChΛi.

Hence ChΛ\Λi → ChΛ by using ChΛi → 0.


Theorem 3.9 (Li and Liu [107], Chance Semicontinuity Law) For eventsΛ1,Λ2, · · · , we have

limi→∞

ChΛi = Ch

limi→∞

Λi

(3.15)

if one of the following conditions is satisfied:(a) ChΛ ≤ 0.5 and Λi ↑ Λ; (b) lim

i→∞ChΛi < 0.5 and Λi ↑ Λ;

(c) ChΛ ≥ 0.5 and Λi ↓ Λ; (d) limi→∞

ChΛi > 0.5 and Λi ↓ Λ.

Proof: (a) Assume ChΛ ≤ 0.5 and Λi ↑ Λ. We first have

ChΛ = supθ∈Θ

(Crθ ∧ PrΛ(θ)), ChΛi = supθ∈Θ

(Crθ ∧ PrΛi(θ))

for i = 1, 2, · · · For each θ ∈ Θ, since Λi(θ) ↑ Λ(θ), it follows from theprobability continuity theorem that

limi→∞

Crθ ∧ PrΛi(θ) = Crθ ∧ PrΛ(θ).

Taking supremum on both sides, we obtain

limi→∞

supθ∈Θ

(Crθ ∧ PrΛi(θ)) = supθ∈Θ

(Crθ ∧ PrΛ(θ).

The part (a) is verified.(b) Assume limi→∞ChΛi < 0.5 and Λi ↑ Λ. For each θ ∈ Θ, since

Crθ ∧ PrΛ(θ) = limi→∞

Crθ ∧ PrΛi(θ),

we have

supθ∈Θ

(Crθ ∧ PrΛ(θ)) ≤ limi→∞

supθ∈Θ

(Crθ ∧ PrΛi(θ)) < 0.5.

It follows that ChΛ < 0.5 and the part (b) holds by using (a).(c) Assume ChΛ ≥ 0.5 and Λi ↓ Λ. We have ChΛc ≤ 0.5 and Λc

i ↑ Λc.It follows from (a) that

limi→∞

ChΛi = 1− limi→∞

ChΛci = 1− ChΛc = ChΛ.

(d) Assume limi→∞ChΛi > 0.5 and Λi ↓ Λ. We have limi→∞

ChΛci <

0.5 and Λci ↑ Λc. It follows from (b) that

limi→∞

ChΛi = 1− limi→∞

ChΛci = 1− ChΛc = ChΛ.



Theorem 3.10 (Chance Asymptotic Theorem) For any events Λ1,Λ2, · · · ,we have

limi→∞

ChΛi ≥ 0.5, if Λi ↑ Θ× Ω, (3.16)

limi→∞

ChΛi ≤ 0.5, if Λi ↓ ∅. (3.17)

Proof: Assume Λi ↑ Θ × Ω. If limi→∞ChΛi < 0.5, it follows from thechance semicontinuity law that

ChΘ× Ω = limi→∞

ChΛi < 0.5

which is in contradiction with CrΘ×Ω = 1. The first inequality is proved.The second one may be verified similarly.

3.2 Hybrid Variables

Recall that a random variable is a measurable function from a probabilityspace to the set of real numbers, and a fuzzy variable is a function from acredibility space to the set of real numbers. In order to describe a quan-tity with both fuzziness and randomness, we introduce a concept of hybridvariable as follows.

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Set of Real Numbers

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•

•

Credibility Space

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RandomVariable

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•

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Probability Space

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FuzzyVariable

HybridVariable

Figure 3.2: Graphical Representation of Hybrid Variable

Definition 3.4 (Liu [136]) A hybrid variable is a measurable function froma chance space (Θ,P,Cr)× (Ω,A,Pr) to the set of real numbers, i.e., for anyBorel set B of real numbers, the set

ξ ∈ B = (θ, ω) ∈ Θ× Ω∣∣ ξ(θ, ω) ∈ B (3.18)

Section 3.2 - Hybrid Variables 125

is an event.

Remark 3.2: A hybrid variable degenerates to a fuzzy variable if the valueof ξ(θ, ω) does not vary with ω. For example,

ξ(θ, ω) = θ, ξ(θ, ω) = θ2 + 1, ξ(θ, ω) = sin θ.

Remark 3.3: A hybrid variable degenerates to a random variable if thevalue of ξ(θ, ω) does not vary with θ. For example,

ξ(θ, ω) = ω, ξ(θ, ω) = ω2 + 1, ξ(θ, ω) = sinω.

Remark 3.4: For each fixed θ ∈ Θ, it is clear that the hybrid variable ξ(θ, ω)is a measurable function from the probability space (Ω,A,Pr) to the set ofreal numbers. Thus it is a random variable and we will denote it by ξ(θ, ·).Then a hybrid variable ξ(θ, ω) may also be regarded as a function from acredibility space (Θ,P,Cr) to the set ξ(θ, ·)|θ ∈ Θ of random variables.Thus ξ is a random fuzzy variable defined by Liu [130].

Remark 3.5: For each fixed ω ∈ Ω, it is clear that the hybrid variableξ(θ, ω) is a function from the credibility space (Θ,P,Cr) to the set of realnumbers. Thus it is a fuzzy variable and we will denote it by ξ(·, ω). Then ahybrid variable ξ(θ, ω) may be regarded as a function from a probability space(Ω,A,Pr) to the set ξ(·, ω)|ω ∈ Ω of fuzzy variables. If Crξ(·, ω) ∈ B isa measurable function of ω for any Borel set B of real numbers, then ξ is afuzzy random variable in the sense of Liu and Liu [150].

Model I

If a is a fuzzy variable and η is a random variable, then the sum ξ = a+ η isa hybrid variable. The product ξ = a · η is also a hybrid variable. Generallyspeaking, if f : <2 → < is a measurable function, then

ξ = f(a, η) (3.19)

is a hybrid variable. Suppose that a has a membership function µ, and η hasa probability density function φ. Then for any Borel set B of real numbers,


we have

Chf(a, η) ∈ B =

supx

(µ(x)

2∧∫

f(x,y)∈B

φ(y)dy

),

if supx

(µ(x)

2∧∫

f(x,y)∈B

φ(y)dy

)< 0.5

1− supx

(µ(x)

2∧∫

f(x,y)∈Bc

φ(y)dy

),

if supx

(µ(x)

2∧∫

f(x,y)∈B

φ(y)dy

)≥ 0.5.

More generally, let a1, a2, · · · , am be fuzzy variables, and let η1, η2, · · · , ηn berandom variables. If f : <m+n → < is a measurable function, then

ξ = f(a1, a2, · · · , am; η1, η2, · · · , ηn) (3.20)

is a hybrid variable. The chance Chf(a1, a2, · · · , am; η1, η2, · · · , ηn) ∈ Bmay be calculated in a similar way provided that µ is the joint membershipfunction and φ is the joint probability density function.

Model II

Let a1, a2, · · · , am be fuzzy variables, and let p1, p2, · · · , pm be nonnegativenumbers with p1 + p2 + · · ·+ pm = 1. Then

ξ =

a1 with probability p1

a2 with probability p2

· · ·am with probability pm

(3.21)

is clearly a hybrid variable. If a1, a2, · · · , am have membership functionsµ1, µ2, · · · , µm, respectively, then for any set B of real numbers, we have

Chξ ∈ B =

supx1,x2··· ,xm

((min

1≤i≤m

µi(xi)2

)∧

m∑i=1

pi |xi ∈ B

),

if supx1,x2··· ,xm

((min

1≤i≤m

µi(xi)2

)∧

m∑i=1

pi |xi ∈ B

)< 0.5

1− supx1,x2··· ,xm

((min

1≤i≤m

µi(xi)2

)∧

m∑i=1

pi |xi ∈ Bc

),

if supx1,x2··· ,xm

((min

1≤i≤m

µi(xi)2

)∧

m∑i=1

pi |xi ∈ B

)≥ 0.5.

Section 3.2 - Hybrid Variables 127

Model III

Let η1, η2, · · · , ηm be random variables, and let u1, u2, · · · , um be nonnegativenumbers with u1 ∨ u2 ∨ · · · ∨ um = 1. Then

ξ =

η1 with membership degree u1

η2 with membership degree u2

· · ·ηm with membership degree um

(3.22)

is clearly a hybrid variable. If η1, η2, · · · , ηm have probability density func-tions φ1, φ2, · · · , φm, respectively, then for any Borel set B of real numbers,we have

Chξ ∈ B =

max1≤i≤m

(ui

2∧∫

B

φi(x)dx),

if max1≤i≤m

(ui

2∧∫

B

φi(x)dx)< 0.5

1− max1≤i≤m

(ui

2∧∫

Bc

φi(x)dx),

if max1≤i≤m

(ui

2∧∫

B

φi(x)dx)≥ 0.5.

Model IV

In many statistics problems, the probability density function is completelyknown except for the values of one or more parameters. For example, itmight be known that the lifetime ξ of a modern engine is an exponentiallydistributed random variable with an unknown expected value β. Usually,there is some relevant information in practice. It is thus possible to specifyan interval in which the value of β is likely to lie, or to give an approximateestimate of the value of β. It is typically not possible to determine the valueof β exactly. If the value of β is provided as a fuzzy variable, then ξ is ahybrid variable. More generally, suppose that ξ has a probability densityfunction

φ(x; a1, a2, · · · , am), x ∈ < (3.23)

in which the parameters a1, a2, · · · , am are fuzzy variables rather than crispnumbers. Then ξ is a hybrid variable provided that φ(x; y1, y2, · · · , ym) isa probability density function for any (y1, y2, · · · , ym) that (a1, a2, · · · , am)may take. If a1, a2, · · · , am have membership functions µ1, µ2, · · · , µm, re-spectively, then for any Borel set B of real numbers, the chance Chξ ∈ B


is

supy1,y2··· ,ym

((min

1≤i≤m

µi(yi)2

)∧∫

B

φ(x; y1, y2, · · · , ym)dx),

if supy1,y2,··· ,ym

((min

1≤i≤m

µi(yi)2

)∧∫

B

φ(x; y1, y2, · · · , ym)dx)< 0.5

1− supy1,y2··· ,ym

((min

1≤i≤m

µi(yi)2

)∧∫

Bc

φ(x; y1, y2, · · · , ym)dx),

if supy1,y2,··· ,ym

((min

1≤i≤m

µi(yi)2

)∧∫

B

φ(x; y1, y2, · · · , ym)dx)≥ 0.5.

When are two hybrid variables equal to each other?

Definition 3.5 Let ξ1 and ξ2 be hybrid variables defined on the chance space(Θ,P,Cr) × (Ω,A,Pr). We say ξ1 = ξ2 if ξ1(θ, ω) = ξ2(θ, ω) for almost all(θ, ω) ∈ Θ× Ω.

Hybrid Vectors

Definition 3.6 An n-dimensional hybrid vector is a measurable functionfrom a chance space (Θ,P,Cr) × (Ω,A,Pr) to the set of n-dimensional realvectors, i.e., for any Borel set B of <n, the set

ξ ∈ B =(θ, ω) ∈ Θ× Ω

∣∣ ξ(θ, ω) ∈ B

(3.24)

is an event.

Theorem 3.11 The vector (ξ1, ξ2, · · · , ξn) is a hybrid vector if and only ifξ1, ξ2, · · · , ξn are hybrid variables.

Proof: Write ξ = (ξ1, ξ2, · · · , ξn). Suppose that ξ is a hybrid vector onthe chance space (Θ,P,Cr) × (Ω,A,Pr). For any Borel set B of <, the setB ×<n−1 is a Borel set of <n. Thus the set

ξ1 ∈ B = ξ1 ∈ B, ξ2 ∈ <, · · · , ξn ∈ < = ξ ∈ B ×<n−1

is an event. Hence ξ1 is a hybrid variable. A similar process may prove thatξ2, ξ3, · · · , ξn are hybrid variables.

Conversely, suppose that all ξ1, ξ2, · · · , ξn are hybrid variables on thechance space (Θ,P,Cr)× (Ω,A,Pr). We define

B =B ⊂ <n

∣∣ ξ ∈ B is an event.

Section 3.3 - Chance Distribution 129

The vector ξ = (ξ1, ξ2, · · · , ξn) is proved to be a hybrid vector if we canprove that B contains all Borel sets of <n. First, the class B contains allopen intervals of <n because

ξ ∈n∏

i=1

(ai, bi)

=

n⋂i=1

ξi ∈ (ai, bi)

is an event. Next, the class B is a σ-algebra of <n because (i) we have <n ∈ Bsince ξ ∈ <n = Θ× Ω; (ii) if B ∈ B, then ξ ∈ B is an event, and

ξ ∈ Bc = ξ ∈ Bc

is an event. This means that Bc ∈ B; (iii) if Bi ∈ B for i = 1, 2, · · · , thenξ ∈ Bi are events and

ξ ∈∞⋃

i=1

Bi

=

∞⋃i=1

ξ ∈ Bi

is an event. This means that ∪iBi ∈ B. Since the smallest σ-algebra con-taining all open intervals of <n is just the Borel algebra of <n, the class Bcontains all Borel sets of <n. The theorem is proved.

Hybrid Arithmetic

Definition 3.7 Let f : <n → < be a measurable function, and ξ1, ξ2, · · · , ξnhybrid variables on the chance space (Θ,P,Cr) × (Ω,A,Pr). Then ξ =f(ξ1, ξ2, · · · , ξn) is a hybrid variable defined as

ξ(θ, ω) = f(ξ1(θ, ω), ξ2(θ, ω), · · · , ξn(θ, ω)), ∀(θ, ω) ∈ Θ× Ω. (3.25)

Example 3.7: Let ξ1 and ξ2 be two hybrid variables. Then the sum ξ =ξ1 + ξ2 is a hybrid variable defined by

ξ(θ, ω) = ξ1(θ, ω) + ξ2(θ, ω), ∀(θ, ω) ∈ Θ× Ω.

The product ξ = ξ1ξ2 is also a hybrid variable defined by

ξ(θ, ω) = ξ1(θ, ω) · ξ2(θ, ω), ∀(θ, ω) ∈ Θ× Ω.

Theorem 3.12 Let ξ be an n-dimensional hybrid vector, and f : <n → < ameasurable function. Then f(ξ) is a hybrid variable.

Proof: Assume that ξ is a hybrid vector on the chance space (Θ,P,Cr) ×(Ω,A,Pr). For any Borel set B of <, since f is a measurable function, thef−1(B) is a Borel set of <n. Thus the set

f(ξ) ∈ B =ξ ∈ f−1(B)

is an event for any Borel set B. Hence f(ξ) is a hybrid variable.


3.3 Chance Distribution

Chance distribution has been defined in several ways. Here we accept thefollowing definition of chance distribution of hybrid variables.

Definition 3.8 (Li and Liu [107]) The chance distribution Φ: < → [0, 1] ofa hybrid variable ξ is defined by

Φ(x) = Ch ξ ≤ x . (3.26)

Example 3.8: Let η be a random variable on a probability space (Ω,A,Pr).It is clear that η may be regarded as a hybrid variable on the chance space(Θ,P,Cr)× (Ω,A,Pr) as follows,

ξ(θ, ω) = η(ω), ∀(θ, ω) ∈ Θ× Ω.

Thus its chance distribution is

Φ(x) = Chξ ≤ x = ChΘ× η ≤ x = CrΘ ∧ Prη ≤ x = Prη ≤ x

which is just the probability distribution of the random variable η.

Example 3.9: Let a be a fuzzy variable on a credibility space (Θ,P,Cr).It is clear that a may be regarded as a hybrid variable on the chance space(Θ,P,Cr)× (Ω,A,Pr) as follows,

ξ(θ, ω) = a(θ), ∀(θ, ω) ∈ Θ× Ω.

Thus its chance distribution is

Φ(x) = Chξ ≤ x = Cha ≤ x × Ω = Cra ≤ x ∧ PrΩ = Cra ≤ x

which is just the credibility distribution of the fuzzy variable a.

Theorem 3.13 (Sufficient and Necessary Condition for Chance Distribu-tion) A function Φ : < → [0, 1] is a chance distribution if and only if it is anincreasing function with

limx→−∞

Φ(x) ≤ 0.5 ≤ limx→+∞

Φ(x), (3.27)

limy↓x

Φ(y) = Φ(x) if limy↓x

Φ(y) > 0.5 or Φ(x) ≥ 0.5. (3.28)

Proof: It is obvious that a chance distribution Φ is an increasing function.The inequalities (3.27) follow from the chance asymptotic theorem immedi-ately. Assume that x is a point at which limy↓x Φ(y) > 0.5. That is,

limy↓x

Chξ ≤ y > 0.5.

Section 3.3 - Chance Distribution 131

Since ξ ≤ y ↓ ξ ≤ x as y ↓ x, it follows from the chance semicontinuitylaw that

Φ(y) = Chξ ≤ y ↓ Chξ ≤ x = Φ(x)

as y ↓ x. When x is a point at which Φ(x) ≥ 0.5, if limy↓x Φ(y) 6= Φ(x), thenwe have

limy↓x

Φ(y) > Φ(x) ≥ 0.5.

For this case, we have proved that limy↓x Φ(y) = Φ(x). Thus (3.28) is proved.Conversely, suppose Φ : < → [0, 1] is an increasing function satisfying

(3.27) and (3.28). Theorem 2.21 states that there is a fuzzy variable whosecredibility distribution is just Φ(x). Since a fuzzy variable is a special hybridvariable, the theorem is proved.

Definition 3.9 The chance density function φ: < → [0,+∞) of a hybridvariable ξ is a function such that

Φ(x) =∫ x

−∞φ(y)dy, ∀x ∈ <, (3.29)

∫ +∞

−∞φ(y)dy = 1 (3.30)

where Φ is the chance distribution of ξ.

Theorem 3.14 Let ξ be a hybrid variable whose chance density function φexists. Then we have

Chξ ≤ x =∫ x

−∞φ(y)dy, Chξ ≥ x =

∫ +∞

x

φ(y)dy. (3.31)

Proof: The first part follows immediately from the definition. In addition,by the self-duality of chance measure, we have

Chξ ≥ x = 1− Chξ < x =∫ +∞

−∞φ(y)dy −

∫ x

−∞φ(y)dy =

∫ +∞

x

φ(y)dy.


Joint Chance Distribution

Definition 3.10 Let (ξ1, ξ2, · · · , ξn) be a hybrid vector. Then the joint chancedistribution Φ : <n → [0, 1] is defined by

Φ(x1, x2, · · · , xn) = Ch ξ1 ≤ x1, ξ2 ≤ x2, · · · , ξn ≤ xn .


Definition 3.11 The joint chance density function φ : <n → [0,+∞) of ahybrid vector (ξ1, ξ2, · · · , ξn) is a function such that

Φ(x1, x2, · · · , xn) =∫ x1

−∞

∫ x2

−∞· · ·∫ xn

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn


−∞

∫ +∞

−∞· · ·∫ +∞

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn = 1

where Φ is the joint chance distribution of the hybrid vector (ξ1, ξ2, · · · , ξn).

3.4 Expected Value

Expected value has been defined in several ways. This book uses the followingdefinition of expected value operator of hybrid variables.

Definition 3.12 (Li and Liu [107]) Let ξ be a hybrid variable. Then theexpected value of ξ is defined by

E[ξ] =∫ +∞

0

Chξ ≥ rdr −∫ 0

−∞Chξ ≤ rdr (3.32)


Example 3.10: If a hybrid variable ξ degenerates to a random variable η,then

Chξ ≤ x = Prη ≤ x, Chξ ≥ x = Prη ≥ x, ∀x ∈ <.

It follows from (3.32) that E[ξ] = E[η]. In other words, the expected valueoperator of hybrid variable coincides with that of random variable.

Example 3.11: If a hybrid variable ξ degenerates to a fuzzy variable a, then

Chξ ≤ x = Cra ≤ x, Chξ ≥ x = Cra ≥ x, ∀x ∈ <.

It follows from (3.32) that E[ξ] = E[a]. In other words, the expected valueoperator of hybrid variable coincides with that of fuzzy variable.

Example 3.12: Let a be a fuzzy variable and η a random variable withfinite expected values. Then the hybrid variable ξ = a+η has expected valueE[ξ] = E[a] + E[η].

Theorem 3.15 Let ξ be a hybrid variable whose chance density function φexists. If the Lebesgue integral ∫ +∞

−∞xφ(x)dx



E[ξ] =∫ +∞

−∞xφ(x)dx. (3.33)


E[ξ] =∫ +∞

0

Chξ ≥ rdr −∫ 0

−∞Chξ ≤ rdr

=∫ +∞

0

[∫ +∞

r

φ(x)dx]

dr −∫ 0

−∞

[∫ r

−∞φ(x)dx

]dr

=∫ +∞

0

[∫ x

0

φ(x)dr]

dx−∫ 0

−∞

[∫ 0

x

φ(x)dr]

dx

=∫ +∞

0

xφ(x)dx+∫ 0

−∞xφ(x)dx

=∫ +∞

−∞xφ(x)dx.


Theorem 3.16 Let ξ be a hybrid variable with chance distribution Φ. If

limx→−∞

Φ(x) = 0, limx→+∞

Φ(x) = 1


−∞xdΦ(x)


E[ξ] =∫ +∞

−∞xdΦ(x). (3.34)


diately have

limy→+∞

∫ y

0

xdΦ(x) =∫ +∞

0


∫ 0

y

xdΦ(x) =∫ 0

−∞xdΦ(x)

and

limy→+∞

∫ +∞

y


∫ y

−∞xdΦ(x) = 0.



y

xdΦ(x) ≥ y

(lim

z→+∞Φ(z)− Φ(y)

)= y (1− Φ(y)) ≥ 0, for y > 0,∫ y

−∞xdΦ(x) ≤ y

(Φ(y)− lim

z→−∞Φ(z)

)= yΦ(y) ≤ 0, for y < 0

thatlim

y→+∞y (1− Φ(y)) = 0, lim

y→−∞yΦ(y) = 0.

Let 0 = x0 < x1 < x2 < · · · < xn = y be a partition of [0, y]. Then we haven−1∑i=0

xi (Φ(xi+1)− Φ(xi)) →∫ y

0

xdΦ(x)

andn−1∑i=0

(1− Φ(xi+1))(xi+1 − xi) →∫ y

0

Chξ ≥ rdr

as max|xi+1 − xi| : i = 0, 1, · · · , n− 1 → 0. Sincen−1∑i=0

xi (Φ(xi+1)− Φ(xi))−n−1∑i=0

(1− Φ(xi+1)(xi+1 − xi) = y(Φ(y)− 1) → 0


0

Chξ ≥ rdr =∫ +∞

0

xdΦ(x).


−∫ 0

−∞Chξ ≤ rdr =

∫ 0

−∞xdΦ(x).


Theorem 3.17 Let ξ be a hybrid variable with finite expected values. Thenfor any real numbers a and b, we have

E[aξ + b] = aE[ξ] + b. (3.35)


E[ξ + b] =∫ +∞

0

Chξ + b ≥ rdr −∫ 0

−∞Chξ + b ≤ rdr

=∫ +∞

0

Chξ ≥ r − bdr −∫ 0

−∞Chξ ≤ r − bdr

= E[ξ] +∫ b

0

(Chξ ≥ r − b+ Chξ < r − b)dr

= E[ξ] + b.



E[aξ + b] = E[ξ]−∫ 0

b

(Chξ ≥ r − b+ Chξ < r − b)dr = E[ξ] + b.

Step 2: We prove E[aξ] = aE[ξ]. If a = 0, then the equation E[aξ] =aE[ξ] holds trivially. If a > 0, we have

E[aξ] =∫ +∞

0

Chaξ ≥ rdr −∫ 0

−∞Chaξ ≤ rdr

=∫ +∞

0

Chξ ≥ r/adr −∫ 0

−∞Chξ ≤ r/adr

= a

∫ +∞

0

Chξ ≥ tdt− a

∫ 0

−∞Chξ ≤ tdt

= aE[ξ].

If a < 0, we have

E[aξ] =∫ +∞

0

Chaξ ≥ rdr −∫ 0

−∞Chaξ ≤ rdr

=∫ +∞

0

Chξ ≤ r/adr −∫ 0

−∞Chξ ≥ r/adr

= a

∫ +∞

0

Chξ ≥ tdt− a

∫ 0

−∞Chξ ≤ tdt

= aE[ξ].

Step 3: For any real numbers a and b, it follows from Steps 1 and 2 that

E[aξ + b] = E[aξ] + b = aE[ξ] + b.


3.5 Variance

Definition 3.13 (Li and Liu [107]) Let ξ be a hybrid variable with finiteexpected value e. Then the variance of ξ is defined by V [ξ] = E[(ξ − e)2].

Theorem 3.18 If ξ is a hybrid variable with finite expected value, a and bare real numbers, then V [aξ + b] = a2V [ξ].


V [aξ + b] = E[(aξ + b− aE[ξ]− b)2

]= a2E[(ξ − E[ξ])2] = a2V [ξ].


Theorem 3.19 Let ξ be a hybrid variable with expected value e. Then V [ξ] =0 if and only if Chξ = e = 1.


E[(ξ − e)2] =∫ +∞

0

Ch(ξ − e)2 ≥ rdr

which implies Ch(ξ − e)2 ≥ r = 0 for any r > 0. Hence we have

Ch(ξ − e)2 = 0 = 1.

That is, Chξ = e = 1.Conversely, if Chξ = e = 1, then we have Ch(ξ − e)2 = 0 = 1 and

Ch(ξ − e)2 ≥ r = 0 for any r > 0. Thus

V [ξ] =∫ +∞

0

Ch(ξ − e)2 ≥ rdr = 0.



Theorem 3.20 Let f be a convex function on [a, b], and ξ a hybrid variablethat takes values in [a, b] and has expected value e. Then

E[f(ξ)] ≤ b− e

b− af(a) +

e− a

b− af(b). (3.36)

Proof: For each (θ, ω) ∈ Θ× Ω, we have a ≤ ξ(θ, ω) ≤ b and

ξ(θ, ω) =b− ξ(θ, ω)b− a

a+ξ(θ, ω)− a

b− ab.


f(ξ(θ, ω)) ≤ b− ξ(θ, ω)b− a

f(a) +ξ(θ, ω)− a

b− af(b).


Theorem 3.21 (Maximum Variance Theorem) Let ξ be a hybrid variablethat takes values in [a, b] and has expected value e. Then

V [ξ] ≤ (e− a)(b− e). (3.37)

Proof: It follows from Theorem 3.20 immediately by defining f(x) = (x−e)2.


3.6 Moments

Definition 3.14 (Li and Liu [107]) Let ξ be a hybrid variable. Then for anypositive integer k,(a) the expected value E[ξk] is called the kth moment;(b) the expected value E[|ξ|k] is called the kth absolute moment;(c) the expected value E[(ξ − E[ξ])k] is called the kth central moment;(d) the expected value E[|ξ−E[ξ]|k] is called the kth absolute central moment.


Theorem 3.22 Let ξ be a nonnegative hybrid variable, and k a positive num-ber. Then the k-th moment

E[ξk] = k

∫ +∞

0

rk−1Chξ ≥ rdr. (3.38)


E[ξk] =∫ ∞

0

Chξk ≥ xdx =∫ ∞

0

Chξ ≥ rdrk = k

∫ ∞

0

rk−1Chξ ≥ rdr.


Theorem 3.23 (Li and Liu [111]) Let ξ be a hybrid variable that takes val-ues in [a, b] and has expected value e. Then for any positive integer k, thekth absolute moment and kth absolute central moment satisfy the followinginequalities,

E[|ξ|k] ≤ b− e

b− a|a|k +

e− a

b− a|b|k, (3.39)

E[|ξ − e|k] ≤ b− e

b− a(e− a)k +

e− a

b− a(b− e)k. (3.40)


3.7 Critical Values

In order to rank hybrid variables, we introduce the following definition ofcritical values of hybrid variables.

Definition 3.15 (Li and Liu [107]) Let ξ be a hybrid variable, and α ∈ (0, 1].Then

ξsup(α) = supr∣∣ Ch ξ ≥ r ≥ α

(3.41)


ξinf(α) = infr∣∣ Ch ξ ≤ r ≥ α

(3.42)



The hybrid variable ξ reaches upwards of the α-optimistic value ξsup(α),and is below the α-pessimistic value ξinf(α) with chance α.

Example 3.13: If a hybrid variable ξ degenerates to a random variable η,then

Chξ ≤ x = Prη ≤ x, Chξ ≥ x = Prη ≥ x, ∀x ∈ <.

It follows from the definition of critical values that

ξsup(α) = ηsup(α), ξinf(α) = ηinf(α), ∀α ∈ (0, 1].

In other words, the critical values of hybrid variable coincide with that ofrandom variable.

Example 3.14: If a hybrid variable ξ degenerates to a fuzzy variable a, then

Chξ ≤ x = Cra ≤ x, Chξ ≥ x = Cra ≥ x, ∀x ∈ <.

It follows from the definition of critical values that

ξsup(α) = asup(α), ξinf(α) = ainf(α), ∀α ∈ (0, 1].

In other words, the critical values of hybrid variable coincide with that offuzzy variable.

Theorem 3.24 Let ξ be a hybrid variable. If α > 0.5, then we have

Chξ ≤ ξinf(α) ≥ α, Chξ ≥ ξsup(α) ≥ α. (3.43)

Proof: It follows from the definition of α-pessimistic value that there existsa decreasing sequence xi such that Chξ ≤ xi ≥ α and xi ↓ ξinf(α) asi→∞. Since ξ ≤ xi ↓ ξ ≤ ξinf(α) and limi→∞Chξ ≤ xi ≥ α > 0.5, itfollows from the chance semicontinuity theorem that

Chξ ≤ ξinf(α) = limi→∞

Chξ ≤ xi ≥ α.

Similarly, there exists an increasing sequence xi such that Chξ ≥xi ≥ α and xi ↑ ξsup(α) as i → ∞. Since ξ ≥ xi ↓ ξ ≥ ξsup(α)and limi→∞Chξ ≥ xi ≥ α > 0.5, it follows from the chance semicontinuitytheorem that

Chξ ≥ ξsup(α) = limi→∞

Chξ ≥ xi ≥ α.


Theorem 3.25 Let ξ be a hybrid variable and α a number between 0 and 1.We have(a) if c ≥ 0, then (cξ)sup(α) = cξsup(α) and (cξ)inf(α) = cξinf(α);(b) if c < 0, then (cξ)sup(α) = cξinf(α) and (cξ)inf(α) = cξsup(α).

Section 3.8 - Distance 139

Proof: (a) If c = 0, then the part (a) is obvious. In the case of c > 0, wehave

(cξ)sup(α) = supr∣∣ Chcξ ≥ r ≥ α

= c supr/c | Chξ ≥ r/c ≥ α

= cξsup(α).

A similar way may prove (cξ)inf(α) = cξinf(α). In order to prove the part (b),it suffices to prove that (−ξ)sup(α) = −ξinf(α) and (−ξ)inf(α) = −ξsup(α).In fact, we have

(−ξ)sup(α) = supr∣∣ Ch−ξ ≥ r ≥ α

= − inf−r | Chξ ≤ −r ≥ α

= −ξinf(α).


Theorem 3.26 Let ξ be a hybrid variable. Then we have(a) if α > 0.5, then ξinf(α) ≥ ξsup(α);(b) if α ≤ 0.5, then ξinf(α) ≤ ξsup(α).


1 ≥ Chξ < ξ(α)+ Chξ > ξ(α) ≥ α+ α > 1.

A contradiction proves ξinf(α) ≥ ξsup(α).Part (b): Assume that ξinf(α) > ξsup(α). It follows from the definition of

ξinf(α) that Chξ ≤ ξ(α) < α. Similarly, it follows from the definition ofξsup(α) that Chξ ≥ ξ(α) < α. Thus

1 ≤ Chξ ≤ ξ(α)+ Chξ ≥ ξ(α) < α+ α ≤ 1.

A contradiction proves ξinf(α) ≤ ξsup(α). The theorem is verified.

Theorem 3.27 Let ξ be a hybrid variable. Then we have(a) ξsup(α) is a decreasing and left-continuous function of α;(b) ξinf(α) is an increasing and left-continuous function of α.

Proof: (a) It is easy to prove that ξinf(α) is an increasing function of α.Next, we prove the left-continuity of ξinf(α) with respect to α. Let αi bean arbitrary sequence of positive numbers such that αi ↑ α. Then ξinf(αi)is an increasing sequence. If the limitation is equal to ξinf(α), then the left-continuity is proved. Otherwise, there exists a number z∗ such that

limi→∞


Thus Chξ ≤ z∗ ≥ αi for each i. Letting i → ∞, we get Chξ ≤ z∗ ≥ α.Hence z∗ ≥ ξinf(α). A contradiction proves the left-continuity of ξinf(α) withrespect to α. The part (b) may be proved similarly.


3.8 Distance

Definition 3.16 (Li and Liu [107]) The distance between hybrid variables ξand η is defined as

d(ξ, η) = E[|ξ − η|]. (3.44)

Theorem 3.28 (Li and Liu [107]) Let ξ, η, τ be hybrid variables, and letd(·, ·) be the distance. Then we have(a) (Nonnegativity) d(ξ, η) ≥ 0;(b) (Identification) d(ξ, η) = 0 if and only if ξ = η;(c) (Symmetry) d(ξ, η) = d(η, ξ);(d) (Triangle Inequality) d(ξ, η) ≤ 2d(ξ, τ) + 2d(η, τ).

Proof: The parts (a), (b) and (c) follow immediately from the definition.Now we prove the part (d). It follows from the chance subadditivity theoremthat

d(ξ, η) =∫ +∞

0

Ch |ξ − η| ≥ rdr

≤∫ +∞

0

Ch |ξ − τ |+ |τ − η| ≥ rdr

≤∫ +∞

0

Ch |ξ − τ | ≥ r/2 ∪ |τ − η| ≥ r/2dr

≤∫ +∞

0

(Ch|ξ − τ | ≥ r/2+ Ch|τ − η| ≥ r/2) dr

=∫ +∞

0

Ch|ξ − τ | ≥ r/2dr +∫ +∞

0

Ch|τ − η| ≥ r/2dr

= 2E[|ξ − τ |] + 2E[|τ − η|] = 2d(ξ, τ) + 2d(τ, η).

3.9 Inequalities

Theorem 3.29 (Li and Liu [107]) Let ξ be a hybrid variable, and f a non-negative function. If f is even and increasing on [0,∞), then for any givennumber t > 0, we have

Ch|ξ| ≥ t ≤ E[f(ξ)]f(t)

. (3.45)


Proof: It is clear that Ch|ξ| ≥ f−1(r) is a monotone decreasing functionof r on [0,∞). It follows from the nonnegativity of f(ξ) that

E[f(ξ)] =∫ +∞

0

Chf(ξ) ≥ rdr

=∫ +∞

0

Ch|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

Ch|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

dr · Ch|ξ| ≥ f−1(f(t))

= f(t) · Ch|ξ| ≥ t


Theorem 3.30 (Li and Liu [107], Markov Inequality) Let ξ be a hybrid vari-able. Then for any given numbers t > 0 and p > 0, we have

Ch|ξ| ≥ t ≤ E[|ξ|p]tp

. (3.46)


Theorem 3.31 (Li and Liu [107], Chebyshev Inequality) Let ξ be a hybridvariable whose variance V [ξ] exists. Then for any given number t > 0, wehave

Ch |ξ − E[ξ]| ≥ t ≤ V [ξ]t2

. (3.47)

Proof: It is a special case of Theorem 3.29 when the hybrid variable ξ isreplaced with ξ − E[ξ], and f(x) = x2.

Theorem 3.32 (Holder’s Inequality) Let p and q be positive real numberswith 1/p + 1/q = 1. If ξ is a fuzzy variable with E[|ξ|p] < ∞ and η is arandom variable with E[|η|q] <∞, then we have

E[|ξη|] ≤ p√E[|ξ|p] q

√E[|η|q]. (3.48)









E[f(|ξ|p, |η|q)] ≤ f(E[|ξ|p], E[|η|q]).


Theorem 3.33 (Minkowski Inequality) Let p be a real number with p ≥ 1.If ξ is a fuzzy variable with E[|ξ|p] < ∞ and η is a random variable withE[|η|p] <∞, then we have

p√E[|ξ + η|p] ≤ p

√E[|ξ|p] + p

√E[|η|p]. (3.49)


√x + p







E[f(|ξ|p, |η|p)] ≤ f(E[|ξ|p], E[|η|p]).


Theorem 3.34 (Jensen’s Inequality) Let ξ be a hybrid variable, and f : < →< a convex function. If E[ξ] and E[f(ξ)] are finite, then

f(E[ξ]) ≤ E[f(ξ)]. (3.50)



f(ξ)− f(E[ξ]) ≥ k · (ξ − E[ξ]).


E[f(ξ)]− f(E[ξ]) ≥ k · (E[ξ]− E[ξ]) = 0




Li and Liu [107] discussed the convergence concepts of hybrid sequence: con-vergence almost surely (a.s.), convergence in chance, convergence in mean,and convergence in distribution.

Table 3.1: Relationship among Convergence Concepts

Convergence⇒

Convergence⇒

Convergence

in Mean in Chance in Distribution

Definition 3.17 Suppose that ξ, ξ1, ξ2, · · · are hybrid variables defined onthe chance space (Θ,P,Cr) × (Ω,A,Pr). The sequence ξi is said to beconvergent a.s. to ξ if there exists an event Λ with ChΛ = 1 such that

limi→∞

|ξi(θ, ω)− ξ(θ, ω)| = 0 (3.51)

for every (θ, ω) ∈ Λ. In that case we write ξi → ξ, a.s.

Definition 3.18 Suppose that ξ, ξ1, ξ2, · · · are hybrid variables. We say thatthe sequence ξi converges in chance to ξ if

limi→∞

Ch |ξi − ξ| ≥ ε = 0 (3.52)

for every ε > 0.

Definition 3.19 Suppose that ξ, ξ1, ξ2, · · · are hybrid variables with finiteexpected values. We say that the sequence ξi converges in mean to ξ if

limi→∞

E[|ξi − ξ|] = 0. (3.53)


limi→∞

E[|ξi − ξ|2] = 0. (3.54)

Definition 3.20 Suppose that Φ,Φ1,Φ2, · · · are the chance distributions ofhybrid variables ξ, ξ1, ξ2, · · · , respectively. We say that ξi converges in dis-tribution to ξ if Φi → Φ at any continuity point of Φ.

Convergence Almost Surely vs. Convergence in Chance

Example 3.15: Convergence a.s. does not imply convergence in chance.Take a credibility space (Θ,P,Cr) to be θ1, θ2, · · · with Crθj = j/(2j+1)


for j = 1, 2, · · · and take an arbitrary probability space (Ω,A,Pr). Then wedefine hybrid variables as

ξi(θj , ω) =

i, if j = i

0, otherwise

for i = 1, 2, · · · and ξ ≡ 0. The sequence ξi convergence a.s. to ξ. However,for some small number ε > 0, we have

Ch|ξi − ξ| ≥ ε = Cr|ξi − ξ| ≥ ε =i

2i+ 1→ 1

2.

That is, the sequence ξi does not converge in chance to ξ.

Example 3.16: Convergence in chance does not imply convergence a.s.Take an arbitrary credibility space (Θ,P,Cr) and take a probability space(Ω,A,Pr) to be [0, 1] with Borel algebra and Lebesgue measure. For anypositive integer i, there is an integer j such that i = 2j + k, where k is aninteger between 0 and 2j − 1. Then we define hybrid variables as

ξi(θ, ω) =

i, if k/2j ≤ ω ≤ (k + 1)/2j

0, otherwise

for i = 1, 2, · · · and ξ ≡ 0. For some small number ε > 0, we have

Ch|ξi − ξ| ≥ ε = Pr|ξi − ξ| ≥ ε =12i→ 0

as i→∞. That is, the sequence ξi converges in chance to ξ. However, forany ω ∈ [0, 1], there is an infinite number of intervals of the form [k/2j , (k +1)/2j ] containing ω. Thus ξi(θ, ω) does converges to 0. In other words, thesequence ξi does not converge a.s. to ξ.

Convergence in Mean vs. Convergence in Chance

Theorem 3.35 Suppose that ξ, ξ1, ξ2, · · · are hybrid variables. If ξi con-verges in mean to ξ, then ξi converges in chance to ξ.

Proof: It follows from the Markov inequality that for any given numberε > 0, we have

Ch|ξi − ξ| ≥ ε ≤ E[|ξi − ξ|]ε

→ 0

as i→∞. Thus ξi converges in chance to ξ. The theorem is proved.

Example 3.17: Convergence in chance does not imply convergence in mean.Take a credibility space (Θ,P,Cr) to be θ1, θ2, · · · with Crθ1 = 1/2


and Crθj = 1/j for j = 2, 3, · · · and take an arbitrary probability space(Ω,A,Pr). The hybrid variables are defined by

ξi(θj , ω) =

i, if j = i

0, otherwise

for i = 1, 2, · · · and ξ ≡ 0. For some small number ε > 0, we have

Ch|ξi − ξ| ≥ ε = Cr|ξi − ξ| ≥ ε =1i→ 0.

That is, the sequence ξi converges in chance to ξ. However,

E[|ξi − ξ|] = 1, ∀i.


Convergence Almost Surely vs. Convergence in Mean

Example 3.18: Convergence a.s. does not imply convergence in mean.Take an arbitrary credibility space (Θ,P,Cr) and take a probability space(Ω,A,Pr) to be ω1, ω2, · · · with Prωj = 1/2j for j = 1, 2, · · · The hybridvariables are defined by

ξi(θ, ωj) =

2i, if j = i

0, otherwise

for i = 1, 2, · · · and ξ ≡ 0. Then ξi converges a.s. to ξ. However, the sequenceξi does not converges in mean to ξ because E[|ξi − ξ|] ≡ 1.

Example 3.19: Convergence in chance does not imply convergence a.s.Take an arbitrary credibility space (Θ,P,Cr) and take a probability space(Ω,A,Pr) to be [0, 1] with Borel algebra and Lebesgue measure. For anypositive integer i, there is an integer j such that i = 2j + k, where k is aninteger between 0 and 2j − 1. The hybrid variables are defined by

ξi(θ, ω) =

i, if k/2j ≤ ω ≤ (k + 1)/2j

0, otherwise

for i = 1, 2, · · · and ξ ≡ 0. Then

E[|ξi − ξ|] =12j→ 0.

That is, the sequence ξi converges in mean to ξ. However, for any ω ∈ [0, 1],there is an infinite number of intervals of the form [k/2j , (k+1)/2j ] containingω. Thus ξi(θ, ω) does converges to 0. In other words, the sequence ξi doesnot converge a.s. to ξ.


Convergence in Chance vs. Convergence in Distribution

Theorem 3.36 Suppose ξ, ξ1, ξ2, · · · are hybrid variables. If ξi convergesin chance to ξ, then ξi converges in distribution to ξ.

Proof: Let x be a given continuity point of the distribution Φ. On the onehand, for any y > x, we have


It follows from the chance subadditivity theorem that

Φi(x) ≤ Φ(y) + Ch|ξi − ξ| ≥ y − x.

Since ξi converges in chance to ξ, we have Ch|ξi − ξ| ≥ y − x → 0 asi → ∞. Thus we obtain lim supi→∞Φi(x) ≤ Φ(y) for any y > x. Lettingy → x, we get

lim supi→∞

Φi(x) ≤ Φ(x). (3.55)


ξ ≤ z = ξi ≤ x, ξ ≤ z ∪ ξi > x, ξ ≤ z ⊂ ξi ≤ x ∪ |ξi − ξ| ≥ x− z

which implies that

Φ(z) ≤ Φi(x) + Ch|ξi − ξ| ≥ x− z.

Since Ch|ξi − ξ| ≥ x − z → 0, we obtain Φ(z) ≤ lim infi→∞Φi(x) for anyz < x. Letting z → x, we get


Φi(x). (3.56)


Example 3.20: Convergence in distribution does not imply convergencein chance. Take a credibility space (Θ,P,Cr) to be θ1, θ2 with Crθ1 =Crθ2 = 1/2 and take an arbitrary probability space (Ω,A,Pr). We definea hybrid variables as

ξ(θ, ω) =

−1, if θ = θ1

1, if θ = θ2.

We also define ξi = −ξ for i = 1, 2, · · · . Then ξi and ξ have the same chancedistribution. Thus ξi converges in distribution to ξ. However, for somesmall number ε > 0, we have

Ch|ξi − ξ| ≥ ε = Cr|ξi − ξ| ≥ ε = 1.

That is, the sequence ξi does not converge in chance to ξ.

Section 3.11 - Conditional Chance 147

Convergence Almost Surely vs. Convergence in Distribution

Example 3.21: Convergence in distribution does not imply convergencea.s. Take a credibility space to be (Θ,P,Cr) to be θ1, θ2 with Crθ1 =Crθ2 = 1/2 and take an arbitrary probability space (Ω,A,Pr). We definea hybrid variable ξ as

ξ(θ, ω) =

−1, if θ = θ1

1, if θ = θ2.

We also define ξi = −ξ for i = 1, 2, · · · . Then ξi and ξ have the same chancedistribution. Thus ξi converges in distribution to ξ. However, the sequenceξi does not converge a.s. to ξ.

Example 3.22: Convergence a.s. does not imply convergence in chance.Take a credibility space (Θ,P,Cr) to be θ1, θ2, · · · with Crθj = j/(2j+1)for j = 1, 2, · · · and take an arbitrary probability space (Ω,A,Pr). The hybridvariables are defined by

ξi(θj , ω) =

i, if j = i

0, otherwise

for i = 1, 2, · · · and ξ ≡ 0. Then the sequence ξi converges a.s. to ξ.However, the chance distributions of ξi are

Φi(x) =

0, if x < 0

(i+ 1)/(2i+ 1), if 0 ≤ x < i

1, if x ≥ i

for i = 1, 2, · · · , respectively. The chance distribution of ξ is

Φ(x) =

0, if x < 01, if x ≥ 0.

It is clear that Φi(x) does not converge to Φ(x) at x > 0. That is, thesequence ξi does not converge in distribution to ξ.

3.11 Conditional Chance

We consider the chance measure of an event A after it has been learned thatsome other event B has occurred. This new chance measure of A is calledthe conditional chance measure of A given B.

In order to define a conditional chance measure ChA|B, at first wehave to enlarge ChA ∩ B because ChA ∩ B < 1 for all events wheneverChB < 1. It seems that we have no alternative but to divide ChA∩B by


ChB. Unfortunately, ChA ∩B/ChB is not always a chance measure.However, the value ChA|B should not be greater than ChA∩B/ChB(otherwise the normality will be lost), i.e.,

ChA|B ≤ ChA ∩BChB

. (3.57)


ChA|B = 1− ChAc|B ≥ 1− ChAc ∩BChB

. (3.58)

Furthermore, since (A∩B)∪ (Ac ∩B) = B, we have ChB ≤ ChA∩B+ChAc ∩B by using the chance subadditivity theorem. Thus

0 ≤ 1− ChAc ∩BChB

≤ ChA ∩BChB

≤ 1. (3.59)

Hence any numbers between 1−ChAc∩B/ChB and ChA∩B/ChBare reasonable values that the conditional chance may take. Based on themaximum uncertainty principle (see Appendix E), we have the following con-ditional chance measure.

Definition 3.21 (Li and Liu [110]) Let (Θ,P,Cr) × (Ω,A,Pr) be a chancespace and A,B two events. Then the conditional chance measure of A givenB is defined by

ChA|B =

ChA ∩BChB

, ifChA ∩B

ChB< 0.5

1− ChAc ∩BChB

, ifChAc ∩B

ChB< 0.5

0.5, otherwise

(3.60)

provided that ChB > 0.

Remark 3.6: It follows immediately from the definition of conditionalchance that

1− ChAc ∩BChB

≤ ChA|B ≤ ChA ∩BChB

. (3.61)

Furthermore, it is clear that the conditional chance measure obeys the max-imum uncertainty principle.


Remark 3.7: Let X and Y be events in the credibility space. Then theconditional chance measure of X × Ω given Y × Ω is

ChX × Ω|Y × Ω =

CrX ∩ Y CrY

, ifCrX ∩ Y

CrY < 0.5

1− CrXc ∩ Y CrY

, ifCrXc ∩ Y

CrY < 0.5

0.5, otherwise

which is just the conditional credibility of X given Y .

Remark 3.8: Let X and Y be events in the probability space. Then theconditional chance measure of Θ×X given Θ× Y is

ChΘ×X|Θ× Y =PrX ∩ Y

PrY

which is just the conditional probability of X given Y .

Example 3.23: Let ξ and η be two hybrid variables. Then we have

Ch ξ = x|η = y=

Chξ = x, η = yChη = y

, ifChξ = x, η = y

Chη = y< 0.5

1− Chξ 6= x, η = yChη = y

, ifChξ 6= x, η = y

Chη = y< 0.5

0.5, otherwise

provided that Chη = y > 0.

Theorem 3.37 (Li and Liu [110]) Conditional chance measure is normal,increasing, self-dual and countably subadditive.

Proof: At first, the conditional chance measure Ch·|B is normal, i.e.,

ChΘ× Ω|B = 1− Ch∅ChB

= 1.


ChA1 ∩BChB

≤ ChA2 ∩BChB

< 0.5,

then

ChA1|B =ChA1 ∩B

ChB≤ ChA2 ∩B

ChB= ChA2|B.

IfChA1 ∩B

ChB≤ 0.5 ≤ ChA2 ∩B

ChB,


then ChA1|B ≤ 0.5 ≤ ChA2|B. If

0.5 <ChA1 ∩B

ChB≤ ChA2 ∩B

ChB,

then we have

ChA1|B =(1− ChAc

1 ∩BChB

)∨0.5 ≤

(1− ChAc

2 ∩BChB

)∨0.5 = ChA2|B.

This means that Ch·|B is increasing. For any event A, if

ChA ∩BChB

≥ 0.5,ChAc ∩B

ChB≥ 0.5,

then we have ChA|B+ChAc|B = 0.5+0.5 = 1 immediately. Otherwise,without loss of generality, suppose

ChA ∩BChB

< 0.5 <ChAc ∩B

ChB,

then we have

ChA|B+ ChAc|B =ChA ∩B

ChB+(

1− ChA ∩BChB

)= 1.

That is, Ch·|B is self-dual. Finally, for any countable sequence Ai ofevents, if ChAi|B < 0.5 for all i, it follows from the countable subadditivityof chance measure that

Ch

∞⋃i=1

Ai ∩B

≤

Ch

∞⋃i=1

Ai ∩B

ChB

≤

∞∑i=1

ChAi ∩B

ChB=

∞∑i=1

ChAi|B.

Suppose there is one term greater than 0.5, say

ChA1|B ≥ 0.5, ChAi|B < 0.5, i = 2, 3, · · ·

If Ch∪iAi|B = 0.5, then we immediately have

Ch

∞⋃i=1

Ai ∩B

≤

∞∑i=1

ChAi|B.

If Ch∪iAi|B > 0.5, we may prove the above inequality by the followingfacts:

Ac1 ∩B ⊂

∞⋃i=2

(Ai ∩B) ∪

( ∞⋂i=1

Aci ∩B

),


ChAc1 ∩B ≤

∞∑i=2

ChAi ∩B+ Ch

∞⋂i=1

Aci ∩B

,

Ch

∞⋃i=1

Ai|B

= 1−

Ch

∞⋂i=1

Aci ∩B

ChB

,

∞∑i=1

ChAi|B ≥ 1− ChAc1 ∩B

ChB+

∞∑i=2

ChAi ∩B

ChB.

If there are at least two terms greater than 0.5, then the countable subad-ditivity is clearly true. Thus Ch·|B is countably subadditive. Hence thetheorem is verified.

Definition 3.22 Let ξ be a hybrid variable on (Θ,P,Cr) × (Ω,A,Pr). Aconditional hybrid variable of ξ given B is a measurable function ξ|B fromthe conditional chance space to the set of real numbers such that

ξ|B(θ, ω) ≡ ξ(θ, ω), ∀(θ, ω) ∈ Θ× Ω. (3.62)

Definition 3.23 (Li and Liu [110]) The conditional chance distribution Φ:< → [0, 1] of a hybrid variable ξ given B is defined by

Φ(x|B) = Ch ξ ≤ x|B (3.63)

provided that ChB > 0.

Example 3.24: Let ξ and η be hybrid variables. Then the conditionalchance distribution of ξ given η = y is

Φ(x|η = y) =

Chξ ≤ x, η = yChη = y

, ifChξ ≤ x, η = y

Chη = y< 0.5

1− Chξ > x, η = yChη = y

, ifChξ > x, η = y

Chη = y< 0.5

0.5, otherwise

provided that Chη = y > 0.

Definition 3.24 (Li and Liu [110]) The conditional chance density functionφ of a hybrid variable ξ given B is a nonnegative function such that

Φ(x|B) =∫ x

−∞φ(y|B)dy, ∀x ∈ <, (3.64)

∫ +∞

−∞φ(y|B)dy = 1 (3.65)

where Φ(x|B) is the conditional chance distribution of ξ given B.


Definition 3.25 (Li and Liu [110]) Let ξ be a hybrid variable. Then theconditional expected value of ξ given B is defined by

E[ξ|B] =∫ +∞

0

Chξ ≥ r|Bdr −∫ 0

−∞Chξ ≤ r|Bdr (3.66)


Following conditional chance and conditional expected value, we also haveconditional variance, conditional moments, conditional critical values, condi-tional entropy as well as conditional convergence.

Chapter 4

Uncertainty Theory

A lot of surveys showed that the measure of union of events is not necessarilythe maximum measure except those events are independent. This fact im-plies that many subjectively uncertain systems do not behave fuzziness andmembership function is no longer helpful. In order to deal with this type ofsubjective uncertainty, Liu [138] founded an uncertainty theory in 2007 thatis a branch of mathematics based on normality, monotonicity, self-duality,countable subadditivity, and product measure axioms.

The emphasis in this chapter is mainly on uncertain measure, uncer-tainty space, uncertain variable, identification function, uncertainty distri-bution, expected value, variance, moments, independence, identical distribu-tion, critical values, entropy, distance, convergence almost surely, convergencein measure, convergence in mean, convergence in distribution, and conditionaluncertainty.

4.1 Uncertainty Space

Let Γ be a nonempty set, and let L be a σ-algebra over Γ. Each element Λ ∈ Lis called an event. In order to present an axiomatic definition of uncertainmeasure, it is necessary to assign to each event Λ a number MΛ whichindicates the level that Λ will occur. In order to ensure that the numberMΛ has certain mathematical properties, Liu [138] proposed the followingfour axioms:

Axiom 1. (Normality) MΓ = 1.

Axiom 2. (Monotonicity) MΛ1 ≤MΛ2 whenever Λ1 ⊂ Λ2.

Axiom 3. (Self-Duality) MΛ+MΛc = 1 for any event Λ.

Axiom 4. (Countable Subadditivity) For every countable sequence of events

154 Chapter 4 - Uncertainty Theory

Λi, we have

M

∞⋃i=1

Λi

≤

∞∑i=1

MΛi. (4.1)

Remark 4.1: The law of contradiction tells us that a proposition cannotbe both true and false at the same time, and the law of excluded middletells us that a proposition is either true or false. Self-duality is in fact ageneralization of the law of contradiction and law of excluded middle. Inother words, a mathematical system without self-duality assumption will beinconsistent with the laws. This is the main reason why self-duality axiom isassumed.

Remark 4.2: Pathology occurs if subadditivity is not assumed. For ex-ample, suppose that a universal set contains 3 elements. We define a setfunction that takes value 0 for each singleton, and 1 for each set with at least2 elements. Then such a set function satisfies all axioms but subadditivity.Is it not strange if such a set function serves as a measure?

Remark 4.3: Pathology occurs if countable subadditivity axiom is replacedwith finite subadditivity axiom. For example, assume the universal set con-sists of all real numbers. We define a set function that takes value 0 if theset is bounded, 0.5 if both the set and complement are unbounded, and 1 ifthe complement of the set is bounded. Then such a set function is finitelysubadditive but not countably subadditive. Is it not strange if such a setfunction serves as a measure? This is the main reason why we accept thecountable subadditivity axiom.

Definition 4.1 (Liu [138]) The set function M is called an uncertain mea-sure if it satisfies the normality, monotonicity, self-duality, and countablesubadditivity axioms.

Example 4.1: Let Γ = γ1, γ2, γ3. For this case, there are only 8 events.Define

Mγ1 = 0.6, Mγ2 = 0.3, Mγ3 = 0.2,

Mγ1, γ2 = 0.8, Mγ1, γ3 = 0.7, Mγ2, γ3 = 0.4,

M∅ = 0, MΓ = 1.

Then M is an uncertain measure because it satisfies the four axioms.

Example 4.2: Suppose that λ(x) is a nonnegative function on < satisfying

supx6=y

(λ(x) + λ(y)) = 1. (4.2)

Section 4.1 - Uncertainty Space 155

Then for any set Λ of real numbers, the set function

MΛ =

supx∈Λ

λ(x), if supx∈Λ

λ(x) < 0.5

1− supx∈Λc

λ(x), if supx∈Λ

λ(x) ≥ 0.5(4.3)

is an uncertain measure on <.

Example 4.3: Suppose ρ(x) is a nonnegative and integrable function on <such that ∫

<ρ(x)dx ≥ 1. (4.4)

Then for any Borel set Λ of real numbers, the set function

MΛ =

∫Λ

ρ(x)dx, if∫

Λ

ρ(x)dx < 0.5

1−∫

Λc

ρ(x)dx, if∫

Λc

ρ(x)dx < 0.5

0.5, otherwise

(4.5)


Example 4.4: Suppose λ(x) is a nonnegative function and ρ(x) is a non-negative and integrable function on < such that

supx∈Λ

λ(x) +∫

Λ

ρ(x)dx ≥ 0.5 and/or supx∈Λc

λ(x) +∫

Λc

ρ(x)dx ≥ 0.5 (4.6)

for any Borel set Λ of real numbers. Then the set function

MΛ =

supx∈Λ

λ(x) +∫

Λ

ρ(x)dx, if supx∈Λ

λ(x) +∫

Λ

ρ(x)dx < 0.5

1− supx∈Λc

λ(x)−∫

Λc

ρ(x)dx, if supx∈Λc

λ(x) +∫

Λc

ρ(x)dx < 0.5

0.5, otherwise


Theorem 4.1 Suppose that M is an uncertain measure. Then we have

M∅ = 0, (4.7)

0 ≤MΛ ≤ 1 (4.8)

for any event Λ.


Proof: It follows from the normality and self-duality axioms that M∅ =1 −MΓ = 1 − 1 = 0. It follows from the monotonicity axiom that 0 ≤MΛ ≤ 1 because ∅ ⊂ Λ ⊂ Γ.

Theorem 4.2 Suppose thatM is an uncertain measure. Then for any eventsΛ1 and Λ2, we have

MΛ1 ∨MΛ2 ≤MΛ1 ∪ Λ2 ≤MΛ1+MΛ2. (4.9)

Proof: The left-hand inequality follows from the monotonicity axiom andthe right-hand inequality follows from the countable subadditivity axiom im-mediately.

Theorem 4.3 Suppose thatM is an uncertain measure. Then for any eventsΛ1 and Λ2, we have

MΛ1+MΛ2 − 1 ≤MΛ1 ∩ Λ2 ≤MΛ1 ∧MΛ2. (4.10)

Proof: The right-hand inequality follows from the monotonicity axiom andthe left-hand inequality follows from the self-duality and countable subaddi-tivity axioms, i.e.,

Mλ1 ∩ Λ2 = 1−M(Λ1 ∩ Λ2)c = 1−MΛc1 ∪ Λc

2

≥ 1− (MΛc1+MΛc

2)

= 1− (1−MΛ1)− (1−MΛ2)

= MΛ1+MΛ2 − 1.

The inequalities are verified.

Theorem 4.4 Let Γ = γ1, γ2, · · · . If M is an uncertain measure, then

Mγi+Mγj ≤ 1 ≤∞∑

k=1

Mγk (4.11)

for any i and j.

Proof: Since M is increasing and self-dual, we have, for any i and j,

Mγi+Mγj ≤MΓ\γj+Mγj = 1.

Since Γ = ∪kγk and M is countably subadditive, we have

1 = MΓ = M

∞⋃k=1

γk

≤

∞∑k=1

Mγk.



Uncertainty Null-Additivity Theorem

Null-additivity is a direct deduction from subadditivity. We first prove amore general theorem.

Theorem 4.5 Let Λi be a decreasing sequence of events with MΛi → 0as i→∞. Then for any event Λ, we have

limi→∞

MΛ ∪ Λi = limi→∞

MΛ\Λi = MΛ. (4.12)

Proof: It follows from the monotonicity and countable subadditivity axiomsthat

MΛ ≤MΛ ∪ Λi ≤MΛ+MΛi

for each i. Thus we get MΛ ∪ Λi → MΛ by using MΛi → 0. Since(Λ\Λi) ⊂ Λ ⊂ ((Λ\Λi) ∪ Λi), we have

MΛ\Λi ≤MΛ ≤MΛ\Λi+MΛi.

Hence MΛ\Λi →MΛ by using MΛi → 0.

Remark 4.4: It follows from the above theorem that the uncertain measureis null-additive, i.e., MΛ1 ∪ Λ2 = MΛ1 + MΛ2 if either MΛ1 = 0or MΛ2 = 0. In other words, the uncertain measure remains unchanged ifthe event is enlarged or reduced by an event with measure zero.

Theorem 4.6 An uncertain measure is identical with not only probabilitymeasure but also credibility measure if there are effectively two elements inthe universal set.

Proof: Suppose there are at most two elements, say γ1 and γ2, takingnonzero uncertain measure values. Let Λ1 and Λ2 be two disjoint events.The argument breaks down into two cases.

Case 1: MΛ1 = 0 or MΛ2 = 0. For this case, we may verify thatMΛ1 ∪Λ2 = MΛ1+MΛ2 by using the countable subadditivity axiom.

Case 2: MΛ1 > 0 or MΛ2 > 0. For this case, without loss of general-ity, we suppose that γ1 ∈ Λ1 and γ2 ∈ Λ2. Note that M(Λ1 ∪ Λ2)c = 0. Itfollows from the self-duality axiom and uncertainty null-additivity theoremthat

MΛ1 ∪ Λ2 = MΛ1 ∪ Λ2 ∪ (Λ1 ∪ Λ2)c = MΓ = 1,

MΛ1+MΛ2 = MΛ1 ∪ (Λ1 ∪ Λ2)c+MΛ2 = 1.

Hence MΛ1∪Λ2 = MΛ1+MΛ2. The additivity is proved. Hence suchan uncertain measure is identical with probability measure. Furthermore, itfollows from Theorem 2.6 that it is also identical with credibility measure.


Uncertainty Asymptotic Theorem

Theorem 4.7 (Uncertainty Asymptotic Theorem) For any events Λ1,Λ2, · · · ,we have

limi→∞

MΛi > 0, if Λi ↑ Γ, (4.13)

limi→∞

MΛi < 1, if Λi ↓ ∅. (4.14)

Proof: Assume Λi ↑ Γ. Since Γ = ∪iΛi, it follows from the countablesubadditivity axioms that

1 = MΓ ≤∞∑

i=1

MΛi.

Since MΛi is increasing with respect to i, we have limi→∞MΛi > 0. IfΛi ↓ ∅, then Λc

i ↑ Γ. It follows from the first inequality and self-duality axiomthat

limi→∞

MΛi = 1− limi→∞

MΛci < 1.


Example 4.5: Assume Γ is the set of real numbers. Let α be a number with0 < α ≤ 0.5. Define a set function as follows,

MΛ =

0, if Λ = ∅α, if Λ is upper bounded0.5, if both Λ and Λc are upper unbounded

1− α, if Λc is upper bounded1, if Λ = Γ.

(4.15)

It is easy to verify that M is an uncertain measure. Write Λi = (−∞, i] fori = 1, 2, · · · Then Λi ↑ Γ and limi→∞MΛi = α. Furthermore, we haveΛc

i ↓ ∅ and limi→∞MΛci = 1− α.

Uncertainty Space

Definition 4.2 (Liu [138]) Let Γ be a nonempty set, L a σ-algebra overΓ, and M an uncertain measure. Then the triplet (Γ,L,M) is called anuncertainty space.

Product Measure Axiom and Product Uncertain Measure

Product uncertain measure was defined by Liu [141] in 2009, thus producingthe fifth axiom of uncertainty theory called product measure axiom. Let(Γk,Lk,Mk) be uncertainty spaces for k = 1, 2, · · · , n. Write

Γ = Γ1 × Γ2 × · · · × Γn, L = L1 × L2 × · · · × Ln.


The product uncertain measure is defined as follows.

Axiom 5. (Liu [141], Product Measure Axiom) Let Γk be nonempty setson which Mk are uncertain measures, k = 1, 2, · · · , n, respectively. Then theproduct uncertain measure on Γ is

MΛ =

supΛ1×Λ2×···×Λn⊂Λ

min1≤k≤n

MkΛk,

if supΛ1×Λ2×···×Λn⊂Λ

min1≤k≤n

MkΛk > 0.5

1− supΛ1×Λ2×···×Λn⊂Λc

min1≤k≤n

MkΛk,

if supΛ1×Λ2×···×Λn⊂Λc

min1≤k≤n

MkΛk > 0.5

0.5, otherwise

(4.16)

for each event Λ ∈ L, denoted by M = M1 ∧M2 ∧ · · · ∧Mn.

Example 4.6: Let Λi be events in (Γk,Lk,Mk), k = 1, 2, · · · , n, respectively.Then the rectangle Λ1×Λ2× · · · ×Λn is an event and the product uncertainmeasure is

MΛ1 × Λ2 × · · · × Λn = min1≤k≤n

MkΛk.

Especially, MΓ = 1 and M∅ = 0.

Theorem 4.8 (Peng [203]) The product uncertain measure (4.16) is an un-certain measure.

Proof: In order to prove that the product uncertain measure (4.16) is indeedan uncertain measure, we should verify that the product uncertain measuresatisfies the normality, monotonicity, self-duality and countable subadditivityaxioms.

Step 1: At first, for any event Λ ∈ L, it is easy to verify that


min1≤k≤n

MkΛk+ supΛ1×Λ2×···×Λn⊂Λc

min1≤k≤n

MkΛk ≤ 1.

This means that at most one of


min1≤k≤n

MkΛk and supΛ1×Λ2×···×Λn⊂Λc

min1≤k≤n

MkΛk

is greater than 0.5. Thus the expression (4.16) is reasonable.

Step 2: The product uncertain measure is clearly normal, i.e.,MΓ = 1.

Step 3: We proves the self-duality, i.e., MΛ+MΛc = 1. The argu-ment breaks down into three cases. Case 1: Assume


min1≤k≤n

MkΛk > 0.5.


Then we immediately have

supΛ1×Λ2×···×Λn⊂Λc

min1≤k≤n

MkΛk < 0.5.

It follows from (4.16) that

MΛ = supΛ1×Λ2×···×Λn⊂Λ

min1≤k≤n

MkΛk,

MΛc = 1− supΛ1×Λ2×···×Λn⊂(Λc)c

min1≤k≤n

MkΛk = 1−MΛ.

The self-duality is proved. Case 2: Assume


min1≤k≤n

MkΛk > 0.5.

This case may be proved by a similar process. Case 3: Assume


min1≤k≤n

MkΛk ≤ 0.5

andsup

Λ1×Λ2×···×Λn⊂Λc

min1≤k≤n

MkΛk ≤ 0.5.

It follows from (4.16) that MΛ = MΛc = 0.5 which proves the self-duality.

Step 4: Let us prove that M is increasing. Suppose Λ and ∆ are twoevents in L with Λ ⊂ ∆. The argument breaks down into three cases. Case 1:Assume


min1≤k≤n

MkΛk > 0.5.

Then

sup∆1×∆2×···×∆n⊂∆

min1≤k≤n

Mk∆k ≥ supΛ1×Λ2×···×Λn⊂Λ

min1≤k≤n

MkΛk > 0.5.

It follows from (4.16) that MΛ ≤M∆. Case 2: Assume

sup∆1×∆2×···×∆n⊂∆c

min1≤k≤n

Mk∆k > 0.5.

Then


min1≤k≤n

MkΛk ≥ sup∆1×∆2×···×∆n⊂∆c

min1≤k≤n

Mk∆k > 0.5.

Thus

MΛ = 1− supΛ1×Λ2×···×Λn⊂Λc

min1≤k≤n

MkΛk

≤ 1− sup∆1×∆2×···×∆n⊂∆c

min1≤k≤n

Mk∆k = M∆.

Section 4.2 - Uncertain Variables 161

Case 3: Assume


min1≤k≤n

MkΛk ≤ 0.5

andsup

∆1×∆2×···×∆n⊂∆c

min1≤k≤n

Mk∆k ≤ 0.5.

ThenMΛ ≤ 0.5 ≤ 1−M∆c = M∆.

Step 5: Finally, we prove the countable subadditivity of M. For sim-plicity, we only prove the case of two events Λ and ∆. The argument breaksdown into three cases. Case 1: Assume MΛ < 0.5 and M∆ < 0.5. Forany given ε > 0, there are two rectangles Λ1 × Λ2 × · · · × Λn ⊂ Λc and∆1 ×∆2 × · · · ×∆n ⊂ ∆c such that

1− min1≤k≤n

MkΛk ≤MΛ+ ε/2,

1− min1≤k≤n

Mk∆k ≤M∆+ ε/2.

Note that

(Λ1 ∩∆1)× (Λ2 ∩∆2)× · · · × (Λn ∩∆n) ⊂ (Λ ∪∆)c.

It follows from Theorem 4.3 that MkΛk ∩ ∆k ≥ MkΛk +Mk∆k − 1for any k. Thus

MΛ ∪∆ ≤ 1− min1≤k≤n

MkΛk ∩∆k

≤ 1− min1≤k≤n

MkΛk+ 1− min1≤k≤n

Mk∆k

≤MΛ+M∆+ ε.

Letting ε → 0, we obtain MΛ ∪ ∆ ≤ MΛ + M∆. Case 2: AssumeMΛ ≥ 0.5 and M∆ < 0.5. When MΛ ∪∆ = 0.5, the subadditivity isobvious. Now we consider the case MΛ∪∆ > 0.5, i.e., MΛc ∩∆c < 0.5.By using Λc ∪∆ = (Λc ∩∆c) ∪∆ and Case 1, we get

MΛc ∪∆ ≤MΛc ∩∆c+M∆.

ThusMΛ ∪∆ = 1−MΛc ∩∆c ≤ 1−MΛc ∪∆+M∆

≤ 1−MΛc+M∆ = MΛ+M∆.

Case 3: If both MΛ ≥ 0.5 and M∆ ≥ 0.5, then the subadditivity isobvious because MΛ+M∆ ≥ 1. The theorem is proved.

Definition 4.3 Let (Γk,Lk,Mk), k = 1, 2, · · · , n be uncertainty spaces, Γ =Γ1 × Γ2 × · · · × Γn, L = L1 × L2 × · · · × Ln, and M = M1 ∧M2 ∧ · · · ∧Mn.Then (Γ,L,M) is called the product uncertainty space of (Γk,Lk,Mk), k =1, 2, · · · , n.


4.2 Uncertain Variables

Definition 4.4 (Liu [138]) An uncertain variable is a measurable functionξ from an uncertainty space (Γ,L,M) to the set of real numbers, i.e., for anyBorel set B of real numbers, the set

ξ ∈ B = γ ∈ Γ∣∣ ξ(γ) ∈ B (4.17)

is an event.

Definition 4.5 An uncertain variable ξ on the uncertainty space (Γ,L,M)is said to be(a) nonnegative if Mξ < 0 = 0;(b) positive if Mξ ≤ 0 = 0;(c) continuous if Mξ = x is a continuous function of x;(d) simple if there exists a finite sequence x1, x2, · · · , xm such that

M ξ 6= x1, ξ 6= x2, · · · , ξ 6= xm = 0; (4.18)


M ξ 6= x1, ξ 6= x2, · · · = 0. (4.19)

Definition 4.6 Let ξ1 and ξ2 be uncertain variables defined on the uncer-tainty space (Γ,L,M). We say ξ1 = ξ2 if ξ1(γ) = ξ2(γ) for almost all γ ∈ Γ.

Uncertain Vector

Definition 4.7 An n-dimensional uncertain vector is a measurable functionfrom an uncertainty space (Γ,L,M) to the set of n-dimensional real vectors,i.e., for any Borel set B of <n, the set

ξ ∈ B =γ ∈ Γ

∣∣ ξ(γ) ∈ B

(4.20)

is an event.

Theorem 4.9 The vector (ξ1, ξ2, · · · , ξn) is an uncertain vector if and onlyif ξ1, ξ2, · · · , ξn are uncertain variables.

Proof: Write ξ = (ξ1, ξ2, · · · , ξn). Suppose that ξ is an uncertain vector onthe uncertainty space (Γ,L,M). For any Borel set B of <, the set B ×<n−1

is a Borel set of <n. Thus the set

ξ1 ∈ B = ξ1 ∈ B, ξ2 ∈ <, · · · , ξn ∈ < = ξ ∈ B ×<n−1

is an event. Hence ξ1 is an uncertain variable. A similar process mayprove that ξ2, ξ3, · · · , ξn are uncertain variables. Conversely, suppose that

Section 4.2 - Uncertain Variables 163

all ξ1, ξ2, · · · , ξn are uncertain variables on the uncertainty space (Γ,L,M).We define

B =B ⊂ <n

∣∣ ξ ∈ B is an event.

The vector ξ = (ξ1, ξ2, · · · , ξn) is proved to be an uncertain vector if we canprove that B contains all Borel sets of <n. First, the class B contains allopen intervals of <n because

ξ ∈n∏

i=1

(ai, bi)

=

n⋂i=1

ξi ∈ (ai, bi)

is an event. Next, the class B is a σ-algebra of <n because (i) we have <n ∈ Bsince ξ ∈ <n = Γ; (ii) if B ∈ B, then ξ ∈ B is an event, and

ξ ∈ Bc = ξ ∈ Bc

is an event. This means that Bc ∈ B; (iii) if Bi ∈ B for i = 1, 2, · · · , thenξ ∈ Bi are events and

ξ ∈∞⋃

i=1

Bi

=

∞⋃i=1

ξ ∈ Bi

is an event. This means that ∪iBi ∈ B. Since the smallest σ-algebra con-taining all open intervals of <n is just the Borel algebra of <n, the class Bcontains all Borel sets of <n. The theorem is proved.

Uncertain Arithmetic

Definition 4.8 Suppose that f : <n → < is a measurable function, andξ1, ξ2, · · · , ξn uncertain variables on the uncertainty space (Γ,L,M). Thenξ = f(ξ1, ξ2, · · · , ξn) is an uncertain variable defined as

ξ(γ) = f(ξ1(γ), ξ2(γ), · · · , ξn(γ)), ∀γ ∈ Γ. (4.21)

Example 4.7: Let ξ1 and ξ2 be two uncertain variables. Then the sumξ = ξ1 + ξ2 is an uncertain variable defined by

ξ(γ) = ξ1(γ) + ξ2(γ), ∀γ ∈ Γ.

The product ξ = ξ1ξ2 is also an uncertain variable defined by

ξ(γ) = ξ1(γ) · ξ2(γ), ∀γ ∈ Γ.

The reader may wonder whether ξ(γ1, γ2, · · · , γn) defined by (4.21) is anuncertain variable. The following theorem answers this question.


Theorem 4.10 Let ξ be an n-dimensional uncertain vector, and f : <n → <a measurable function. Then f(ξ) is an uncertain variable such that

Mf(ξ) ∈ B = Mξ ∈ f−1(B) (4.22)

for any Borel set B of real numbers.

Proof: Assume that ξ is an uncertain vector on the uncertainty space(Γ,L,M). For any Borel set B of <, since f is a measurable function, thef−1(B) is a Borel set of <n. Thus the set f(ξ) ∈ B = ξ ∈ f−1(B) is anevent for any Borel set B. Hence f(ξ) is an uncertain variable.

Example 4.8: Let ξ be a simple uncertain variable defined by

ξ =

x1 with uncertain measure µ1

x2 with uncertain measure µ2

· · ·xm with uncertain measure µm

where x1, x2, · · · , xm are distinct numbers. If f is a one-to-one function, thenf(ξ) is also a simple uncertain variable

f(ξ) =

f(x1) with uncertain measure µ1

f(x2) with uncertain measure µ2

· · ·f(xm) with uncertain measure µm.

4.3 Identification Function

A random variable may be characterized by a probability density function,and a fuzzy variable may be described by a membership function. Thissection will introduce an identification function to characterize an uncertainvariable.

First Identification Function

Definition 4.9 An uncertain variable ξ is said to have a first identificationfunction λ if(i) λ(x) is a nonnegative function on < such that

supx6=y

(λ(x) + λ(y)) = 1; (4.23)

(ii) for any set B of real numbers, we have

Mξ ∈ B =

supx∈B

λ(x), if supx∈B

λ(x) < 0.5

1− supx∈Bc

λ(x), if supx∈B

λ(x) ≥ 0.5.(4.24)

Section 4.3 - Identification Function 165

Example 4.9: By a rectangular uncertain variable we mean the uncertainvariable fully determined by the pair (a, b) of crisp numbers with a < b, whosefirst identification function is

λ1(x) =

0.5, if a ≤ x ≤ b

0, otherwise.

Example 4.10: By a triangular uncertain variable we mean the uncertainvariable fully determined by the triplet (a, b, c) of crisp numbers with a <b < c, whose first identification function is

λ2(x) =

x− a

2(b− a), if a ≤ x ≤ b

x− c

2(b− c), if b ≤ x ≤ c

0, otherwise.

Example 4.11: By a trapezoidal uncertain variable we mean the uncertainvariable fully determined by the quadruplet (a, b, c, d) of crisp numbers witha < b < c < d, whose first identification function is

λ3(x) =

x− a

2(b− a), if a ≤ x ≤ b

0.5, if b ≤ x ≤ c

x− d

2(c− d), if c ≤ x ≤ d

0, otherwise.

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x

λ1(x)

0

0.5

a b

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λ2(x)

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λ3(x)

a b c d.................................................................................................................................................................................................................................................................................................................................................................................................

..

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..

..

..

.

Figure 4.1: First Identification Functions λ1, λ2 and λ3

Theorem 4.11 Suppose λ is a nonnegative function satisfying (4.23). Thenthere is an uncertain variable ξ such that (4.24) holds.


Proof: Let < be the universal set. For each set B of real numbers, we definea set function

MB =

supx∈B

λ(x), if supx∈B

λ(x) < 0.5

1− supx∈Bc

λ(x), if supx∈B

λ(x) ≥ 0.5.

It is clear that M is normal, increasing, self-dual, and countably subadditive.That is, the set function M is indeed an uncertain measure. Now we definean uncertain variable ξ as an identity function from the uncertainty space(<,A,M) to <. We may verify that ξ meets (4.24). The theorem is proved.

Theorem 4.12 (First Measure Inversion Theorem) Let ξ be an uncertainvariable with first identification function λ. Then for any set B of real num-bers, we have

Mξ ∈ B =

supx∈B

λ(x), if supx∈B

λ(x) < 0.5

1− supx∈Bc

λ(x), if supx∈B

λ(x) ≥ 0.5.(4.25)

Proof: It follows from the definition of first identification function immedi-ately.

Example 4.12: Suppose ξ is an uncertain variable with first identificationfunction λ. Then

Mξ = x = λ(x), ∀x ∈ <.

Example 4.13: Suppose ξ is a discrete uncertain variable taking values inx1, x2, · · · with first identification function λ. Note that λ must meet

supi 6=j

(λ(xi) + λ(xj)) = 1. (4.26)

For any set B of real numbers, we have

Mξ ∈ B =

supxi∈B

λ(xi), if supxi∈B

λ(xi) < 0.5

1− supxi∈Bc

λ(xi), if supxi∈B

λ(xi) ≥ 0.5.(4.27)

Second Identification Function

Definition 4.10 (Liu [147]) An uncertain variable ξ is said to have a secondidentification function ρ if(i) ρ(x) is a nonnegative and integrable function on < such that∫

<ρ(x)dx ≥ 1; (4.28)


(ii) for any Borel set B of real numbers, we have

Mξ ∈ B =

∫B

ρ(x)dx, if∫

B

ρ(x)dx < 0.5

1−∫

Bc

ρ(x)dx, if∫

Bc

ρ(x)dx < 0.5

0.5, otherwise.

(4.29)

Example 4.14: By an exponential uncertain variable we mean the uncertainvariable whose second identification function is

ρ1(x) =

1β

exp(−xα

), if x ≥ 0

0, if x < 0(4.30)

where α and β are real numbers with α ≥ β > 0. Note that∫ +∞

−∞ρ1(x)dx =

α

β≥ 1.

Example 4.15: By a bell-like uncertain variable we mean the uncertainvariable whose second identification function is

ρ2(x) =1

β√π

exp(− (x− µ)2

α2

), x ∈ < (4.31)

where µ, α and β are real numbers with α ≥ β > 0. Note that∫ +∞

−∞ρ2(x)dx =

α

β≥ 1.

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ρ1(x)

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ρ2(x)................................................................................................................................................................................................................................................................................................................... ......................

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µ................................

Figure 4.2: Second Identification Functions ρ1 and ρ2


Theorem 4.13 Suppose ρ is a nonnegative and integrable function satisfying(4.28). Then there is an uncertain variable ξ such that (4.29) holds.

Proof: Let < be the universal set. For each Borel set B of real numbers, wedefine a set function

MB =

∫B

ρ(x)dx, if∫

B

ρ(x)dx < 0.5

1−∫

Bc

ρ(x)dx, if∫

Bc

ρ(x)dx < 0.5

0.5, otherwise.


Theorem 4.14 (Second Measure Inversion Theorem) Let ξ be an uncertainvariable with second identification function ρ. Then for any Borel set B ofreal numbers, we have

Mξ ∈ B =

∫B

ρ(x)dx, if∫

B

ρ(x)dx < 0.5

1−∫

Bc

ρ(x)dx, if∫

Bc

ρ(x)dx < 0.5

0.5, otherwise.

(4.32)

Proof: It follows from the definition of second identification function imme-diately.

Example 4.16: Suppose ξ is a discrete uncertain variable taking values inx1, x2, · · · with second identification function ρ. Note that ρ must meet

∞∑i=1

ρ(xi) ≥ 1. (4.33)

For any set B of real numbers, we have

Mξ ∈ B =

∑xi∈B

ρ(xi), if∑

xi∈B

ρ(xi) < 0.5

1−∑

xi∈Bc

ρ(xi), if∑

xi∈Bc

ρ(xi) < 0.5

0.5, otherwise.

(4.34)


Third Identification Function

Definition 4.11 (Liu [141]) An uncertain variable ξ is said to have a thirdidentification function (λ, ρ) if (i) λ(x) is a nonnegative function and ρ(x) isa nonnegative and integrable function such that

supx∈B

λ(x)+∫

B

ρ(x)dx ≥ 0.5 and/or supx∈Bc

λ(x)+∫

Bc

ρ(x)dx ≥ 0.5 (4.35)

for any Borel set B of real numbers; (ii) we have

Mξ ∈ B =

supx∈B

λ(x) +∫

B

ρ(x)dx,

if supx∈B

λ(x) +∫

B

ρ(x)dx < 0.5

1− supx∈Bc

λ(x)−∫

Bc

ρ(x)dx,

if supx∈Bc

λ(x) +∫

Bc

ρ(x)dx < 0.5

0.5, otherwise.

(4.36)

Theorem 4.15 Suppose λ is a nonnegative function and ρ is a nonnegativeand integrable function satisfying the condition (4.35). Then there is anuncertain variable ξ such that (4.36) holds.

Proof: Let < be the universal set. For each Borel set B of real numbers, wedefine a set function

MB =

supx∈B

λ(x) +∫

B

ρ(x)dx, if supx∈B

λ(x) +∫

B

ρ(x)dx < 0.5

1− supx∈Bc

λ(x)−∫

Bc

ρ(x)dx, if supx∈Bc

λ(x) +∫

Bc

ρ(x)dx < 0.5

0.5, otherwise.


Theorem 4.16 (Third Measure Inversion Theorem) Let ξ be an uncertainvariable with third identification function (λ, ρ). Then for any Borel set B of


real numbers, we have

Mξ ∈ B =

supx∈B

λ(x) +∫

B

ρ(x)dx,

if supx∈B

λ(x) +∫

B

ρ(x)dx < 0.5

1− supx∈Bc

λ(x)−∫

Bc

ρ(x)dx,

if supx∈Bc

λ(x) +∫

Bc

ρ(x)dx < 0.5

0.5, otherwise.

Proof: It follows from the definition of third identification function imme-diately.

Example 4.17: Suppose ξ is a discrete uncertain variable taking values inx1, x2, · · · with third identification function (λ, ρ). Assume that

supxi∈B

λ(xi) +∑

xi∈B

ρ(xi) ≥ 0.5 and/or supxi∈Bc

λ(xi) +∑

xi∈Bc

ρ(xi) ≥ 0.5

for any set B of real numbers. Then we have

Mξ ∈ B =

supxi∈B

λ(xi) +∑

xi∈B

ρ(xi),

if supxi∈B

λ(xi) +∑

xi∈B

ρ(xi) < 0.5

1− supxi∈Bc

λ(xi)−∑

xi∈Bc

ρ(xi),

if supxi∈Bc

λ(xi) +∑

xi∈Bc

ρ(xi) < 0.5

0.5, otherwise.

4.4 Uncertainty Distribution

Following the idea of probability distribution, credibility distribution andchance distribution, this section introduces uncertainty distribution in orderto describe uncertain variables.

Definition 4.12 (Liu [138]) The uncertainty distribution Φ: < → [0, 1] ofan uncertain variable ξ is defined by

Φ(x) = M ξ ≤ x . (4.37)

Section 4.4 - Uncertainty Distribution 171

Example 4.18: Let ξ be an uncertain variable with first identification func-tion λ. Then its uncertainty distribution is

Φ(x) =

supy≤x

λ(y), if supy≤x

λ(y) < 0.5

1− supy>x

λ(y), if supy≤x

λ(y) ≥ 0.5.

Example 4.19: Let ξ be an uncertain variable with second identificationfunction ρ. Then its uncertainty distribution is

Φ(x) =

∫ x

−∞ρ(y)dy, if

∫ x

−∞ρ(y)dy < 0.5

1−∫ +∞

x

ρ(y)dy, if∫ +∞

x

ρ(y)dy < 0.5

0.5, otherwise.

Example 4.20: Let ξ be an uncertain variable with third identificationfunction (λ, ρ). Then its uncertainty distribution is

Φ(x) =

supy≤x

λ(y) +∫ x

−∞ρ(y)dy,

if supy≤x

λ(y) +∫ x

−∞ρ(y)dy < 0.5

1− supy>x

λ(y)−∫ +∞

x

ρ(y)dy,

if supy>x

λ(y) +∫ +∞

x

ρ(y)dy < 0.5

0.5, otherwise.

Theorem 4.17 An uncertainty distribution is an increasing function suchthat

0 ≤ limx→−∞

Φ(x) < 1, 0 < limx→+∞

Φ(x) ≤ 1. (4.38)

Proof: It is obvious that an uncertainty distribution Φ is an increasingfunction, and the inequalities (4.38) follow from the uncertainty asymptotictheorem immediately.

Example 4.21: Let c be a number with 0 < c < 1. Then Φ(x) ≡ c is anuncertainty distribution. When c ≤ 0.5, we define a set function over < as


follows,

MΛ =

0, if Λ = ∅c, if Λ is upper bounded

0.5, if both Λ and Λc are upper unbounded1− c, if Λc is upper bounded

1, if Λ = Γ.

Then (<,L,M) is an uncertainty space. It is easy to verify that the identityfunction ξ(γ) = γ is an uncertain variable whose uncertainty distribution isjust Φ(x) ≡ c. When c > 0.5, we define

MΛ =

0, if Λ = ∅1− c, if Λ is upper bounded0.5, if both Λ and Λc are upper unboundedc, if Λc is upper bounded1, if Λ = Γ.

Then the function ξ(γ) = −γ is an uncertain variable whose uncertaintydistribution is just Φ(x) ≡ c.

Definition 4.13 A continuous uncertain variable is said to be (a) singularif its uncertainty distribution is a singular function; (b) absolutely continuousif its uncertainty distribution is absolutely continuous.

Definition 4.14 (Liu [138]) The uncertainty density function φ: < → [0,+∞)of an uncertain variable ξ is a function such that

Φ(x) =∫ x

−∞φ(y)dy, ∀x ∈ <, (4.39)

∫ +∞

−∞φ(y)dy = 1 (4.40)

where Φ is the uncertainty distribution of ξ.

Example 4.22: The uncertainty density function may not exist even if theuncertainty distribution is continuous and differentiable a.e. Suppose f is theCantor function, and set

Φ(x) =

0, if x < 0

f(x), if 0 ≤ x ≤ 11, if x > 1.

(4.41)


Then Φ is an increasing and continuous function, and is an uncertainty dis-tribution. Note that Φ′(x) = 0 almost everywhere, and∫ +∞

−∞Φ′(x)dx = 0 6= 1.

Thus the uncertainty density function does not exist.

Theorem 4.18 Let ξ be an uncertain variable whose uncertainty densityfunction φ exists. Then we have

Mξ ≤ x =∫ x

−∞φ(y)dy, Mξ ≥ x =

∫ +∞

x

φ(y)dy. (4.42)

Proof: The first part follows immediately from the definition. In addition,by the self-duality of uncertain measure, we have

Mξ ≥ x = 1−Mξ < x =∫ +∞

−∞φ(y)dy −

∫ x

−∞φ(y)dy =

∫ +∞

x

φ(y)dy.


Joint Uncertainty Distribution

Definition 4.15 Let (ξ1, ξ2, · · · , ξn) be an uncertain vector. Then the jointuncertainty distribution Φ : <n → [0, 1] is defined by

Φ(x1, x2, · · · , xn) = M ξ1 ≤ x1, ξ2 ≤ x2, · · · , ξn ≤ xn .

Definition 4.16 The joint uncertainty density function φ : <n → [0,+∞)of an uncertain vector (ξ1, ξ2, · · · , ξn) is a function such that

Φ(x1, x2, · · · , xn) =∫ x1

−∞

∫ x2

−∞· · ·∫ xn

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn


−∞

∫ +∞

−∞· · ·∫ +∞

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn = 1

where Φ is the joint uncertainty distribution of (ξ1, ξ2, · · · , ξn).

4.5 Independence

Definition 4.17 (Liu [141]) The uncertain variables ξ1, ξ2, · · · , ξm are saidto be independent if

M

m⋂

i=1

ξi ∈ Bi

= min

1≤i≤mM ξi ∈ Bi (4.43)

for any Borel sets B1, B2, · · · , Bm of real numbers.


Theorem 4.18 The uncertain variables ξ1, ξ2, · · · , ξm are independent if andonly if

M

m⋃

i=1

ξi ∈ Bi

= max

1≤i≤mM ξi ∈ Bi (4.44)

for any Borel sets B1, B2, · · · , Bm of real numbers.

Proof: It follows from the self-duality of uncertain measure that ξ1, ξ2, · · · , ξmare independent if and only if

M

m⋃

i=1

ξi ∈ Bi

= 1−M

m⋂

i=1

ξi ∈ Bci

= 1− min

1≤i≤mMξi ∈ Bc

i = max1≤i≤m

M ξi ∈ Bi .

Thus the proof is complete.


This section introduces the concept of identical distribution of uncertain vari-ables.

Definition 4.19 The uncertain variables ξ and η are identically distributedif

Mξ ∈ B = Mη ∈ B (4.45)

for any Borel set B of real numbers.

Theorem 4.19 Let ξ and η be identically distributed uncertain variables,and f : < → < a measurable function. Then f(ξ) and f(η) are identicallydistributed uncertain variables.

Proof: For any Borel set B of real numbers, we have

Mf(ξ) ∈ B = Mξ ∈ f−1(B) = Mη ∈ f−1(B) = Mf(η) ∈ B.

Hence f(ξ) and f(η) are identically distributed uncertain variables.

Theorem 4.20 If ξ and η are identically distributed uncertain variables,then they have the same uncertainty distribution.

Proof: Since ξ and η are identically distributed uncertain variables, we haveMξ ∈ (−∞, x] = Mη ∈ (−∞, x] for any x. Thus ξ and η have the sameuncertainty distribution.

Theorem 4.21 If ξ and η are identically distributed uncertain variableswhose uncertainty density functions exist, then they have the same uncer-tainty density function.

Proof: It follows from Theorem 4.20 immediately.

Section 4.7 - Operational Law 175

4.7 Operational Law

Theorem 4.22 (Liu [141], Operational Law) Let ξ1, ξ2, · · · , ξn be indepen-dent uncertain variables, and f : <n → < a measurable function. Thenξ = f(ξ1, ξ2, · · · , ξn) is an uncertain variable such that

Mξ ∈ B =

supf(B1,B2,··· ,Bn)⊂B

min1≤k≤n

Mkξk ∈ Bk,

if supf(B1,B2,··· ,Bn)⊂B

min1≤k≤n

Mkξk ∈ Bk > 0.5

1− supf(B1,B2,··· ,Bn)⊂Bc

min1≤k≤n

Mkξk ∈ Bk,

if supf(B1,B2,··· ,Bn)⊂Bc

min1≤k≤n

Mkξk ∈ Bk > 0.5

0.5, otherwise

where B,B1, B2, · · · , Bn are Borel sets of real numbers.

Proof: It follows from the product measure axiom immediately.

Example 4.23: The sum of independent rectangular uncertain variablesξ = (a1, a2) and η = (b1, b2) is a rectangular uncertain variable, and

ξ + η = (a1 + b1, a2 + b2).

The product of a rectangular uncertain variable ξ = (a1, a2) and a scalarnumber h is also a rectangular uncertain variable

h · ξ =

(ha1, ha2), if h ≥ 0

(ha2, ha1), if h < 0.

Example 4.24: The sum of independent triangular uncertain variables ξ =(a1, a2, a3) and η = (b1, b2, b3) is a triangular uncertain variable, and

ξ + η = (a1 + b1, a2 + b2, a3 + b3).

The product of a triangular uncertain variable ξ = (a1, a2, a3) and a scalarnumber h is also a triangular uncertain variable

h · ξ =

(ha1, ha2, ha3), if h ≥ 0

(ha3, ha2, ha1), if h < 0.

Example 4.25: The sum of independent trapezoidal uncertain variablesξ = (a1, a2, a3, a4) and η = (b1, b2, b3, b4) is a trapezoidal uncertain variable,and

ξ + η = (a1 + b1, a2 + b2, a3 + b3, a4 + b4).


The product of a trapezoidal uncertain variable ξ = (a1, a2, a3, a4) and ascalar number h is also a trapezoidal uncertain variable

h · ξ =

(ha1, ha2, ha3, ha4), if h ≥ 0

(ha4, ha3, ha2, ha1), if h < 0.

Example 4.26: Let ξ1, ξ2, · · · , ξn be independent uncertain variables withfirst identification functions λ1, λ2, · · · , λn, respectively, and f : <n → < afunction. It follows from the operational law that ξ = f(ξ1, ξ2, · · · , ξn) is anuncertain variable with first identification function

λ(x) =

sup

f(x1,··· ,xn)=x

min1≤i≤n

λi(xi), if supf(x1,··· ,xn)=x

min1≤i≤n

λi(xi) < 0.5

1− supf(x1,··· ,xn) 6=x

min1≤i≤n

λi(xi), if supf(x1,··· ,xn)=x

min1≤i≤n

λi(xi) ≥ 0.5.

Example 4.27: Let ξ and η be two independent exponential uncertainvariables whose second identification function is

ρξ(x) =

1a

exp(−xa

), if x ≥ 0

0, if x < 0,ρη(x) =

1b

exp(−xb

), if x ≥ 0

0, if x < 0.

Then their sum is also an exponential uncertain variable with second identi-fication function

ρ(x) =

1

a+ bexp

(− x

a+ b

), if x ≥ 0

0, if x < 0.

4.8 Expected Value

Expected value is the average value of uncertain variable in the sense ofuncertain measure, and represents the size of uncertain variable.

Definition 4.20 (Liu [138]) Let ξ be an uncertain variable. Then the ex-pected value of ξ is defined by

E[ξ] =∫ +∞

0

Mξ ≥ rdr −∫ 0

−∞Mξ ≤ rdr (4.46)


Example 4.28: The rectangular uncertain variable ξ = (a, b) has an ex-pected value (a+ b)/2.


Example 4.29: The triangular uncertain variable ξ = (a, b, c) has an ex-pected value E[ξ] = (a+ 2b+ c)/4.

Example 4.30: The trapezoidal uncertain variable ξ = (a, b, c, d) has anexpected value E[ξ] = (a+ b+ c+ d)/4.

Example 4.31: Let ξ be a continuous uncertain variable with first iden-tification function λ. If its expected value exists, and there is a point x0

such that λ(x) is increasing on (−∞, x0) and decreasing on (x0,+∞), thenMξ ≥ x = λ(x) for any x > x0 and Mξ ≤ x = λ(x) for any x < x0.Thus

E[ξ] = x0 +∫ +∞

x0

λ(x)dx−∫ x0

−∞λ(x)dx.

Example 4.32: The definition of expected value operator is also applicableto discrete case. Assume that ξ is a simple uncertain variable whose firstidentification function is

λ(x) =

λ1, if x = x1

λ2, if x = x2

· · ·λm, if x = xm

(4.47)

where x1, x2, · · · , xm are distinct numbers. The expected value of ξ is

E[ξ] =m∑

i=1

wixi (4.48)


wi =

max

1≤j≤mλj |xj ≤ xi − max

1≤j≤mλj |xj < xi

+ max1≤j≤m

λj |xj ≥ xi − max1≤j≤m

λj |xj > xi, if λi < 0.5

1− max1≤j≤m

λj |xj < xi − max1≤j≤m

λj |xj > xi, if λi ≥ 0.5

for i = 1, 2, · · · ,m. It is easy to verify that all wi ≥ 0 and the sum of allweights is just 1.

Example 4.33: Consider the uncertain variable ξ defined by (4.47). Supposex1 < x2 < · · · < xm and there exists an index k with 1 < k < m such that

λ1 ≤ λ2 ≤ · · · ≤ λk and λk ≥ λk+1 ≥ · · · ≥ λm.


Then the expected value is determined by (4.48) and the weights are givenby

wi =

λ1, if i = 1λi − λi−1, if i = 2, 3, · · · , k − 1

1− λk−1 − λk+1, if i = k

λi − λi+1, if i = k + 1, k + 2, · · · ,m− 1λm, if i = m.

Example 4.34: The exponential uncertain variable ξ with second identifi-cation function

ρ(x) =

1β

exp(−xα

), if x ≥ 0

0, if x < 0

has an expected value

E[ξ] = α+α(β − α)

βln

2α2α− β

+α

2ln

2α− β

β.

Example 4.35: The bell-like uncertain variable ξ with second identificationfunction

ρ(x) =1

β√π

exp(− (x− µ)2

α2

)has an expected value E[ξ] = µ.

Example 4.36: Let ξ be a continuous uncertain variable with second iden-tification function ρ. It is clear that there are two points x1 and x2 suchthat ∫ x1

−∞ρ(x)dx =

∫ +∞

x2

ρ(x)dx = 0.5.

If the expected value exists, then we have

E[ξ] =∫ x1

−∞xρ(x)dx+

∫ +∞

x2

xρ(x)dx.

Example 4.37: Assume that ξ is a simple uncertain variable whose secondidentification function is

ρ(x) =

ρ1, if x = x1

ρ2, if x = x2

· · ·ρm, if x = xm

(4.49)



E[ξ] =m∑

i=1

wixi (4.50)


wi =

ρi, if

∑xj≤xi

ρj < 0.5 or∑

xj≥xi

ρj < 0.5

(0.5−

∑xj<xi

ρi

)+

+(0.5−

∑xj>xi

ρi

)+

, otherwise

for i = 1, 2, · · · ,m.

Example 4.38: Assume that ξ is a simple uncertain variable whose thirdidentification function is

(λ, ρ)(x) =

(λ1, ρ1), if x = x1

(λ1, ρ2), if x = x2

· · ·(λm, ρm), if x = xm

(4.51)


E[ξ] =m∑

i=1

wixi (4.52)


wi =

supxj≤xi

λj − supxj<xi

λj + ρi, if supxj≤xi

λj +∑

xj≤xi

ρi < 0.5

supxj≥xi

λj − supxj>xi

λj + ρi, if supxj≥xi

λj +∑

xj≥xi

ρi < 0.5

(0.5− sup

xj<xi

λj −∑

xj<xi

ρi

)+

+(0.5− sup

xj>xi

λj −∑

xj>xi

ρi

)+

, otherwise

for i = 1, 2, · · · ,m.

Theorem 4.23 Let ξ be an uncertain variable whose uncertainty densityfunction φ exists. If the Lebesgue integral∫ +∞

−∞xφ(x)dx


E[ξ] =∫ +∞

−∞xφ(x)dx. (4.53)



E[ξ] =∫ +∞

0

Mξ ≥ rdr −∫ 0

−∞Mξ ≤ rdr

=∫ +∞

0

[∫ +∞

r

φ(x)dx]

dr −∫ 0

−∞

[∫ r

−∞φ(x)dx

]dr

=∫ +∞

0

[∫ x

0

φ(x)dr]

dx−∫ 0

−∞

[∫ 0

x

φ(x)dr]

dx

=∫ +∞

0

xφ(x)dx+∫ 0

−∞xφ(x)dx

=∫ +∞

−∞xφ(x)dx.


Theorem 4.24 Let ξ be an uncertain variable with uncertainty distributionΦ. If

limx→−∞

Φ(x) = 0, limx→∞

Φ(x) = 1


−∞xdΦ(x)


E[ξ] =∫ +∞

−∞xdΦ(x). (4.54)


diately have

limy→+∞

∫ y

0

xdΦ(x) =∫ +∞

0


∫ 0

y

xdΦ(x) =∫ 0

−∞xdΦ(x)

and

limy→+∞

∫ +∞

y


∫ y

−∞xdΦ(x) = 0.


y

xdΦ(x) ≥ y

(lim

z→+∞Φ(z)− Φ(y)

)= y (1− Φ(y)) ≥ 0, for y > 0,


∫ y

−∞xdΦ(x) ≤ y

(Φ(y)− lim

z→−∞Φ(z)

)= yΦ(y) ≤ 0, for y < 0

thatlim

y→+∞y (1− Φ(y)) = 0, lim

y→−∞yΦ(y) = 0.


n−1∑i=0

xi (Φ(xi+1)− Φ(xi)) →∫ y

0

xdΦ(x)

andn−1∑i=0

(1− Φ(xi+1))(xi+1 − xi) →∫ y

0

Mξ ≥ rdr


n−1∑i=0

xi (Φ(xi+1)− Φ(xi))−n−1∑i=0

(1− Φ(xi+1)(xi+1 − xi) = y(Φ(y)− 1) → 0


0

Mξ ≥ rdr =∫ +∞

0

xdΦ(x).


−∫ 0

−∞Mξ ≤ rdr =

∫ 0

−∞xdΦ(x).


Theorem 4.25 Let ξ be an uncertain variable with finite expected value.Then for any real numbers a and b, we have

E[aξ + b] = aE[ξ] + b. (4.55)


E[ξ + b] =∫ +∞

0

Mξ + b ≥ rdr −∫ 0

−∞Mξ + b ≤ rdr

=∫ +∞

0

Mξ ≥ r − bdr −∫ 0

−∞Mξ ≤ r − bdr

= E[ξ] +∫ b

0

(Mξ ≥ r − b+Mξ < r − b)dr

= E[ξ] + b.



E[aξ + b] = E[ξ]−∫ 0

b

(Mξ ≥ r − b+Mξ < r − b)dr = E[ξ] + b.

Step 2: We prove E[aξ] = aE[ξ]. If a = 0, then the equation E[aξ] =aE[ξ] holds trivially. If a > 0, we have

E[aξ] =∫ +∞

0

Maξ ≥ rdr −∫ 0

−∞Maξ ≤ rdr

=∫ +∞

0

Mξ ≥ r/adr −∫ 0

−∞Mξ ≤ r/adr

= a

∫ +∞

0

Mξ ≥ tdt− a

∫ 0

−∞Mξ ≤ tdt

= aE[ξ].

If a < 0, we have

E[aξ] =∫ +∞

0

Maξ ≥ rdr −∫ 0

−∞Maξ ≤ rdr

=∫ +∞

0

Mξ ≤ r/adr −∫ 0

−∞Mξ ≥ r/adr

= a

∫ +∞

0

Mξ ≥ tdt− a

∫ 0

−∞Mξ ≤ tdt

= aE[ξ].

Step 3: Finally, for any real numbers a and b, it follows from Steps 1and 2 that

E[aξ + b] = E[aξ] + b = aE[ξ] + b.


Theorem 4.26 Let f be a convex function on [a, b], and ξ an uncertainvariable that takes values in [a, b] and has expected value e. Then

E[f(ξ)] ≤ b− e

b− af(a) +

e− a

b− af(b). (4.56)

Proof: For each γ ∈ Γ, we have a ≤ ξ(γ) ≤ b and

ξ(γ) =b− ξ(γ)b− a

a+ξ(γ)− a

b− ab.


f(ξ(γ)) ≤ b− ξ(γ)b− a

f(a) +ξ(γ)− a

b− af(b).

Taking expected values on both sides, we obtain the inequality.


4.9 Variance

The variance of an uncertain variable provides a measure of the spread of thedistribution around its expected value. A small value of variance indicatesthat the uncertain variable is tightly concentrated around its expected value;and a large value of variance indicates that the uncertain variable has a widespread around its expected value.

Definition 4.21 (Liu [138]) Let ξ be an uncertain variable with finite ex-pected value e. Then the variance of ξ is defined by V [ξ] = E[(ξ − e)2].

Example 4.39: Let ξ be a rectangular uncertain variable (a, b). Then itsexpected value is e = (a+ b)/2, and for any positive number r, we have

M(ξ − e)2 ≥ r =

1/2, if r ≤ (b− a)2/4

0, if r > (b− a)2/4.

Thus the variance is

V [ξ] =∫ +∞

0

M(ξ − e)2 ≥ rdr =∫ (b−a)2/4

0

12dr =

(b− a)2

8.

Example 4.40: Let ξ = (a, b, c) be a triangular uncertain variable. Thenits variance is

V [ξ] =33α3 + 21α2β + 11αβ2 − β3

384αwhere α = (b − a) ∨ (c − b) and β = (b − a) ∧ (c − b). Especially, if ξ issymmetric, i.e., b− a = c− b, then its variance is V [ξ] = (c− a)2/24.

Example 4.41: Let ξ = (a, b, c, d) be a symmetric trapezoidal uncertainvariable, i.e., b− a = d− c. Then its variance is

V [ξ] =(d− a)2 + (d− a)(c− b) + (c− b)2

24.

Example 4.42: An uncertain variable ξ is called normal if it has a normaluncertainty distribution

Φ(x) =(

1 + exp(π(e− x)√

3σ

))−1

, x ∈ <. (4.57)

Note that the uncertain variable with first identification function

λ(x) =

(1 + exp

(π(e− x)√

3σ

))−1

, if x ≤ e

(1 + exp

(π(x− e)√

3σ

))−1

, if x > e


.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...............

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

................

...............

x

Φ(x)

0

1

0.5

e.......................................................................................

......................................................

..................................

............................................................................................................................................................

.....................................

.............................................................

...................................................................................................................................................

..

..

..

..

..

..

..

..

...........................................................................

Figure 4.3: Normal Uncertainty Distribution

has the normal uncertainty distribution Φ; and the uncertain variable withsecond identification function

ρ(x) =πx√3σ

(2 + exp

(π(e− x)√

3σ

)+ exp

(π(x− e)√

3σ

))−1

has also the normal uncertainty distribution Φ. It is easy to verify thatthe normal uncertain variable ξ has expected value e. It follows from thedefinition of variance that

V [ξ] =∫ +∞

0

M(ξ − e)2 ≥ rdr =∫ +∞

0

M(ξ ≥ e+√r)∪ (ξ ≤ e−

√r)dr.

On the one hand, we have

V [ξ] ≥∫ +∞

0

Mξ ≥ e+√rdr =

∫ +∞

0

(1 + exp

(π√r√

3σ

))−1

dr =σ2

2.

On the other hand, we have

V [ξ] ≤∫ +∞

0

(Mξ ≥ e+√r+Mξ ≤ e−

√r)dr

= 2∫ +∞

0

(1 + exp

(π√r√

3σ

))−1

dr = σ2.

This means that the variance of a normal uncertain variable is an interval[σ2/2, σ2]. If a scalar variance is needed, then we take the maximum value,i.e.,

V [ξ] = σ2. (4.58)

Let ξ1 and ξ2 be independent normal uncertain variables with expected valuese1 and e2, variances σ2

1 and σ22 , respectively. Then for any real numbers a1

and a2, the uncertain variable a1ξ1 +a2ξ2 is also normal with expected valuea1e1 + a2e2 and variance (|a1|σ1 + |a2|σ2)2.


Theorem 4.27 If ξ is an uncertain variable with finite expected value, a andb are real numbers, then V [aξ + b] = a2V [ξ].


V [aξ + b] = E[(aξ + b− aE[ξ]− b)2

]= a2E[(ξ − E[ξ])2] = a2V [ξ].

Theorem 4.28 Let ξ be an uncertain variable with expected value e. ThenV [ξ] = 0 if and only if Mξ = e = 1.


E[(ξ − e)2] =∫ +∞

0

M(ξ − e)2 ≥ rdr

which implies M(ξ − e)2 ≥ r = 0 for any r > 0. Hence we have

M(ξ − e)2 = 0 = 1.

That is, Mξ = e = 1.Conversely, if Mξ = e = 1, then we have M(ξ − e)2 = 0 = 1 and

M(ξ − e)2 ≥ r = 0 for any r > 0. Thus

V [ξ] =∫ +∞

0

M(ξ − e)2 ≥ rdr = 0.


Theorem 4.29 Let ξ be an uncertain variable that takes values in [a, b] andhas expected value e. Then

V [ξ] ≤ (e− a)(b− e). (4.59)

Proof: It follows from Theorem 4.26 immediately by defining f(x) = (x−e)2.

4.10 Moments

Definition 4.22 (Liu [138]) Let ξ be an uncertain variable. Then for anypositive integer k,(a) the expected value E[ξk] is called the kth moment;(b) the expected value E[|ξ|k] is called the kth absolute moment;(c) the expected value E[(ξ − E[ξ])k] is called the kth central moment;(d) the expected value E[|ξ−E[ξ]|k] is called the kth absolute central moment.



Theorem 4.30 Let ξ be a nonnegative uncertain variable, and k a positivenumber. Then the k-th moment

E[ξk] = k

∫ +∞

0

rk−1Mξ ≥ rdr. (4.60)


E[ξk] =∫ ∞

0

Mξk ≥ xdx =∫ ∞

0

Mξ ≥ rdrk = k

∫ ∞

0

rk−1Mξ ≥ rdr.


Theorem 4.31 Let ξ be an uncertain variable, and t a positive number. IfE[|ξ|t] <∞, then

limx→∞

xtM|ξ| ≥ x = 0. (4.61)

Conversely, if (4.61) holds for some positive number t, then E[|ξ|s] <∞ forany 0 ≤ s < t.

Proof: It follows from the definition of expected value operator that

E[|ξ|t] =∫ +∞

0

M|ξ|t ≥ rdr <∞.

Thus we have

limx→∞

∫ +∞

xt/2

M|ξ|t ≥ rdr = 0.

The equation (4.61) is proved by the following relation,∫ +∞

xt/2

M|ξ|t ≥ rdr ≥∫ xt

xt/2

M|ξ|t ≥ rdr ≥ 12xtM|ξ| ≥ x.

Conversely, if (4.61) holds, then there exists a number a such that

xtM|ξ| ≥ x ≤ 1, ∀x ≥ a.

Thus we have

E[|ξ|s] =∫ a

0

M|ξ|s ≥ rdr +∫ +∞

a

M|ξ|s ≥ rdr

=∫ a

0

M|ξ|s ≥ rdr +∫ +∞

a

srs−1M|ξ| ≥ rdr

≤∫ a

0

M|ξ|s ≥ rdr + s

∫ +∞

0

rs−t−1dr

< +∞.

(by∫ +∞

0

rpdr <∞ for any p < −1)



Theorem 4.32 Let ξ be an uncertain variable that takes values in [a, b] andhas expected value e. Then for any positive integer k, the kth absolute momentand kth absolute central moment satisfy the following inequalities,

E[|ξ|k] ≤ b− e

b− a|a|k +

e− a

b− a|b|k, (4.62)

E[|ξ − e|k] ≤ b− e

b− a(e− a)k +

e− a

b− a(b− e)k. (4.63)


4.11 Critical Values

Definition 4.23 (Liu [138]) Let ξ be an uncertain variable, and α ∈ (0, 1].Then

ξsup(α) = supr∣∣M ξ ≥ r ≥ α

(4.64)


ξinf(α) = infr∣∣M ξ ≤ r ≥ α

(4.65)


Theorem 4.33 Let ξ be an uncertain variable and α a number between 0and 1. We have(a) if c ≥ 0, then (cξ)sup(α) = cξsup(α) and (cξ)inf(α) = cξinf(α);(b) if c < 0, then (cξ)sup(α) = cξinf(α) and (cξ)inf(α) = cξsup(α).

Proof: (a) If c = 0, then the part (a) is obvious. In the case of c > 0, wehave

(cξ)sup(α) = supr∣∣Mcξ ≥ r ≥ α

= c supr/c | Mξ ≥ r/c ≥ α

= cξsup(α).

A similar way may prove (cξ)inf(α) = cξinf(α). In order to prove the part (b),it suffices to prove that (−ξ)sup(α) = −ξinf(α) and (−ξ)inf(α) = −ξsup(α).In fact, we have

(−ξ)sup(α) = supr∣∣M−ξ ≥ r ≥ α

= − inf−r | Mξ ≤ −r ≥ α

= −ξinf(α).



Theorem 4.34 Let ξ be an uncertain variable. Then we have(a) if α > 0.5, then ξinf(α) ≥ ξsup(α);(b) if α ≤ 0.5, then ξinf(α) ≤ ξsup(α).


1 ≥Mξ < ξ(α)+Mξ > ξ(α) ≥ α+ α > 1.

A contradiction proves ξinf(α) ≥ ξsup(α).Part (b): Assume that ξinf(α) > ξsup(α). It follows from the definition

of ξinf(α) that Mξ ≤ ξ(α) < α. Similarly, it follows from the definition ofξsup(α) that Mξ ≥ ξ(α) < α. Thus

1 ≤Mξ ≤ ξ(α)+Mξ ≥ ξ(α) < α+ α ≤ 1.

A contradiction proves ξinf(α) ≤ ξsup(α). The theorem is verified.

Theorem 4.35 Let ξ be an uncertain variable. Then ξsup(α) is a decreasingfunction of α, and ξinf(α) is an increasing function of α.

Proof: It follows from the definition immediately.

4.12 Entropy

This section provides a definition of entropy to characterize the uncertaintyof uncertain variables resulting from information deficiency.

Entropy of Discrete Uncertain Variables

Definition 4.24 (Liu [138]) Suppose that ξ is a discrete uncertain variabletaking values in x1, x2, · · · . Then its entropy is defined by

H[ξ] =∞∑

i=1

S(Mξ = xi) (4.66)


Example 4.43: Suppose that ξ is a discrete uncertain variable taking valuesin x1, x2, · · · . If there exists some index k such that Mξ = xk = 1, and0 otherwise, then its entropy H[ξ] = 0.

Example 4.44: Suppose that ξ is a simple uncertain variable taking valuesin x1, x2, · · · , xn. If Mξ = xi = 0.5 for all i = 1, 2, · · · , n, then itsentropy H[ξ] = n ln 2.


Theorem 4.36 Suppose that ξ is a discrete uncertain variable taking valuesin x1, x2, · · · . Then

H[ξ] ≥ 0 (4.67)

and equality holds if and only if ξ is essentially a deterministic/crisp number.

Proof: The nonnegativity is clear. In addition, H[ξ] = 0 if and only ifMξ = xi = 0 or 1 for each i. That is, there exists one and only one index ksuch that Mξ = xk = 1, i.e., ξ is essentially a deterministic/crisp number.

This theorem states that the entropy of an uncertain variable reaches itsminimum 0 when the uncertain variable degenerates to a deterministic/crispnumber. In this case, there is no uncertainty.

Theorem 4.37 Suppose that ξ is a simple uncertain variable taking valuesin x1, x2, · · · , xn. Then

H[ξ] ≤ n ln 2 (4.68)

and equality holds if and only if Mξ = xi = 0.5 for all i = 1, 2, · · · , n.

Proof: Since the function S(t) reaches its maximum ln 2 at t = 0.5, we have

H[ξ] =n∑

i=1

S(Mξ = xi) ≤ n ln 2

and equality holds if and only if Mξ = xi = 0.5 for all i = 1, 2, · · · , n.

This theorem states that the entropy of an uncertain variable reaches itsmaximum when the uncertain variable is an equipossible one. In this case,there is no preference among all the values that the uncertain variable willtake.

Entropy of Continuous Uncertain Variables

Definition 4.25 (Liu [141]) Suppose that ξ is a continuous uncertain vari-able. Then its entropy is defined by

H[ξ] =∫ +∞

−∞S(Mξ ≤ x)dx (4.69)


Example 4.45: Let ξ be a continuous uncertain variable with uncertaintydistribution

Φ(x) =

0, if x < a

0.5, if a ≤ x < b

1, if x ≥ b.

(4.70)


Its entropy is

H[ξ] = −∫ b

a

(0.5 ln 0.5 + (1− 0.5) ln (1− 0.5)) dx = (b− a) ln 2.

Theorem 4.38 Let ξ be a continuous uncertain variable. Then H[ξ] > 0.

Proof: The positivity is clear. In addition, when a continuous uncertainvariable tends to a crisp number, its entropy tends to the minimum 0. How-ever, a crisp number is not a continuous uncertain variable.

Theorem 4.39 Let ξ be a continuous uncertain variable taking values onthe interval [a, b]. Then

H[ξ] ≤ (b− a) ln 2 (4.71)

and equality holds if and only if ξ has an uncertainty distribution (4.70).

Proof: The theorem follows from the fact that the function S(t) reaches itsmaximum ln 2 at t = 0.5.

4.13 Distance

Definition 4.26 (Liu [138]) The distance between uncertain variables ξ andη is defined as

d(ξ, η) = E[|ξ − η|]. (4.72)

Example 4.46: Let ξ and η be rectangular uncertain variables (a1, b1) and(a2, b2), respectively, and (a1, b1)∩ (a2, b2) = ∅. Then |ξ− η| is a rectangularuncertain variable on the interval with endpoints |a1−b2| and |b1−a2|. Thusthe distance between ξ and η is the expected value of |ξ − η|, i.e.,

d(ξ, η) =12

(|a1 − b2|+ |b1 − a2|) .

Example 4.47: Let ξ = (a1, b1, c1) and η = (a2, b2, c2) be triangular uncer-tain variables such that (a1, c1) ∩ (a2, c2) = ∅. Then

d(ξ, η) =14

(|a1 − c2|+ 2|b1 − b2|+ |c1 − a2|) .

Example 4.48: Let ξ = (a1, b1, c1, d1) and η = (a2, b2, c2, d2) be trapezoidaluncertain variables such that (a1, d1) ∩ (a2, d2) = ∅. Then

d(ξ, η) =14

(|a1 − d2|+ |b1 − c2|+ |c1 − b2|+ |d1 − a2|) .


Theorem 4.40 Let ξ, η, τ be uncertain variables, and let d(·, ·) be the dis-tance. Then we have(a) (Nonnegativity) d(ξ, η) ≥ 0;(b) (Identification) d(ξ, η) = 0 if and only if ξ = η;(c) (Symmetry) d(ξ, η) = d(η, ξ);(d) (Triangle Inequality) d(ξ, η) ≤ 2d(ξ, τ) + 2d(η, τ).

Proof: The parts (a), (b) and (c) follow immediately from the definition.Now we prove the part (d). It follows from the countable subadditivity axiomthat

d(ξ, η) =∫ +∞

0

M |ξ − η| ≥ rdr

≤∫ +∞

0

M |ξ − τ |+ |τ − η| ≥ rdr

≤∫ +∞

0

M |ξ − τ | ≥ r/2 ∪ |τ − η| ≥ r/2dr

≤∫ +∞

0

(M|ξ − τ | ≥ r/2+M|τ − η| ≥ r/2) dr

=∫ +∞

0

M|ξ − τ | ≥ r/2dr +∫ +∞

0

M|τ − η| ≥ r/2dr

= 2E[|ξ − τ |] + 2E[|τ − η|] = 2d(ξ, τ) + 2d(τ, η).

Example 4.49: Let Γ = γ1, γ2, γ3. Define M∅ = 0, MΓ = 1 andMΛ = 1/2 for any subset Λ (excluding ∅ and Γ). We set uncertain variablesξ, η and τ as follows,

ξ(γ) =

1, if γ 6= γ3

0, otherwise,η(γ) =

−1, if γ 6= γ1

0, otherwise,τ(γ) ≡ 0.

It is easy to verify that d(ξ, τ) = d(τ, η) = 1/2 and d(ξ, η) = 3/2. Thus

d(ξ, η) =32(d(ξ, τ) + d(τ, η)).

4.14 Inequalities

Theorem 4.41 (Liu [138]) Let ξ be an uncertain variable, and f a non-negative function. If f is even and increasing on [0,∞), then for any givennumber t > 0, we have

M|ξ| ≥ t ≤ E[f(ξ)]f(t)

. (4.73)


Proof: It is clear that M|ξ| ≥ f−1(r) is a monotone decreasing functionof r on [0,∞). It follows from the nonnegativity of f(ξ) that

E[f(ξ)] =∫ +∞

0

Mf(ξ) ≥ rdr

=∫ +∞

0

M|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

M|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

dr ·M|ξ| ≥ f−1(f(t))

= f(t) ·M|ξ| ≥ t


Theorem 4.42 (Liu [138], Markov Inequality) Let ξ be an uncertain vari-able. Then for any given numbers t > 0 and p > 0, we have

M|ξ| ≥ t ≤ E[|ξ|p]tp

. (4.74)


Theorem 4.43 (Liu [138], Chebyshev Inequality) Let ξ be an uncertain vari-able whose variance V [ξ] exists. Then for any given number t > 0, we have

M |ξ − E[ξ]| ≥ t ≤ V [ξ]t2

. (4.75)

Proof: It is a special case of Theorem 4.41 when the uncertain variable ξ isreplaced with ξ − E[ξ], and f(x) = x2.

Theorem 4.44 (Liu [138], Holder’s Inequality) Let p and q be positive realnumbers with 1/p+1/q = 1, and let ξ and η be strongly independent uncertainvariables with E[|ξ|p] <∞ and E[|η|q] <∞. Then we have

E[|ξη|] ≤ p√E[|ξ|p] q

√E[|η|q]. (4.76)









E[f(|ξ|p, |η|q)] ≤ f(E[|ξ|p], E[|η|q]).


Theorem 4.45 (Liu [138], Minkowski Inequality) Let p be a real numberwith p ≥ 1, and let ξ and η be strongly independent uncertain variables withE[|ξ|p] <∞ and E[|η|p] <∞. Then we have

p√E[|ξ + η|p] ≤ p

√E[|ξ|p] + p

√E[|η|p]. (4.77)


√x + p







E[f(|ξ|p, |η|p)] ≤ f(E[|ξ|p], E[|η|p]).


Theorem 4.46 (Liu [138], Jensen’s Inequality) Let ξ be an uncertain vari-able, and f : < → < a convex function. If E[ξ] and E[f(ξ)] are finite, then

f(E[ξ]) ≤ E[f(ξ)]. (4.78)



f(ξ)− f(E[ξ]) ≥ k · (ξ − E[ξ]).


E[f(ξ)]− f(E[ξ]) ≥ k · (E[ξ]− E[ξ]) = 0




We have the following four convergence concepts of uncertain sequence: con-vergence almost surely (a.s.), convergence in measure, convergence in mean,and convergence in distribution.

Table 4.1: Relationship among Convergence Concepts

Convergence⇒

Convergence⇒

Convergence

in Mean in Measure in Distribution

Definition 4.27 (Liu [138]) Suppose that ξ, ξ1, ξ2, · · · are uncertain vari-ables defined on the uncertainty space (Γ,L,M). The sequence ξi is saidto be convergent a.s. to ξ if there exists an event Λ with MΛ = 1 such that

limi→∞

|ξi(γ)− ξ(γ)| = 0 (4.79)

for every γ ∈ Λ. In that case we write ξi → ξ, a.s.

Definition 4.28 (Liu [138]) Suppose that ξ, ξ1, ξ2, · · · are uncertain vari-ables. We say that the sequence ξi converges in measure to ξ if

limi→∞

M |ξi − ξ| ≥ ε = 0 (4.80)

for every ε > 0.

Definition 4.29 (Liu [138]) Suppose that ξ, ξ1, ξ2, · · · are uncertain vari-ables with finite expected values. We say that the sequence ξi converges inmean to ξ if

limi→∞

E[|ξi − ξ|] = 0. (4.81)


limi→∞

E[|ξi − ξ|2] = 0. (4.82)

Definition 4.30 (Liu [138]) Suppose that Φ,Φ1,Φ2, · · · are the uncertaintydistributions of uncertain variables ξ, ξ1, ξ2, · · · , respectively. We say thatξi converges in distribution to ξ if Φi → Φ at any continuity point of Φ.

Theorem 4.31 (Liu [138]) Suppose that ξ, ξ1, ξ2, · · · are uncertain variables.If ξi converges in mean to ξ, then ξi converges in measure to ξ.

Proof: It follows from the Markov inequality that for any given numberε > 0, we have

M|ξi − ξ| ≥ ε ≤ E[|ξi − ξ|]ε

→ 0

as i→∞. Thus ξi converges in measure to ξ. The theorem is proved.

Section 4.16 - Conditional Uncertainty 195

Theorem 4.32 (Liu [138]) Suppose ξ, ξ1, ξ2, · · · are uncertain variables. Ifξi converges in measure to ξ, then ξi converges in distribution to ξ.

Proof: Let x be a given continuity point of the uncertainty distribution Φ.On the one hand, for any y > x, we have


It follows from the countable subadditivity axiom that

Φi(x) ≤ Φ(y) +M|ξi − ξ| ≥ y − x.

Since ξi converges in measure to ξ, we have M|ξi − ξ| ≥ y − x → 0 asi → ∞. Thus we obtain lim supi→∞Φi(x) ≤ Φ(y) for any y > x. Lettingy → x, we get

lim supi→∞

Φi(x) ≤ Φ(x). (4.83)


ξ ≤ z = ξi ≤ x, ξ ≤ z ∪ ξi > x, ξ ≤ z ⊂ ξi ≤ x ∪ |ξi − ξ| ≥ x− z

which implies that

Φ(z) ≤ Φi(x) +M|ξi − ξ| ≥ x− z.

Since M|ξi − ξ| ≥ x − z → 0, we obtain Φ(z) ≤ lim infi→∞Φi(x) for anyz < x. Letting z → x, we get


Φi(x). (4.84)


4.16 Conditional Uncertainty

We consider the uncertain measure of an event A after it has been learnedthat some other event B has occurred. This new uncertain measure of A iscalled the conditional uncertain measure of A given B.

In order to define a conditional uncertain measure MA|B, at first wehave to enlarge MA ∩ B because MA ∩ B < 1 for all events wheneverMB < 1. It seems that we have no alternative but to divide MA∩B byMB. Unfortunately, MA∩B/MB is not always an uncertain measure.However, the value MA|B should not be greater than MA ∩ B/MB(otherwise the normality will be lost), i.e.,

MA|B ≤ MA ∩BMB

. (4.85)



MA|B = 1−MAc|B ≥ 1− MAc ∩BMB

. (4.86)

Furthermore, since (A ∩B) ∪ (Ac ∩B) = B, we have MB ≤MA ∩B+MAc ∩B by using the countable subadditivity axiom. Thus

0 ≤ 1− MAc ∩BMB

≤ MA ∩BMB

≤ 1. (4.87)

Hence any numbers between 1−MAc∩B/MB andMA∩B/MB arereasonable values that the conditional uncertain measure may take. Based onthe maximum uncertainty principle (see Appendix E), we have the followingconditional uncertain measure.

Definition 4.33 (Liu [138]) Let (Γ,L,M) be an uncertainty space, and A,B ∈L. Then the conditional uncertain measure of A given B is defined by

MA|B =

MA ∩BMB

, ifMA ∩BMB

< 0.5

1− MAc ∩BMB

, ifMAc ∩BMB

< 0.5

0.5, otherwise

(4.88)

provided that MB > 0.

It follows immediately from the definition of conditional uncertain mea-sure that

1− MAc ∩BMB

≤MA|B ≤ MA ∩BMB

. (4.89)

Furthermore, the conditional uncertain measure obeys the maximum uncer-tainty principle, and takes values as close to 0.5 as possible.

Remark 4.5: Conditional uncertain measure coincides with conditionalprobability, conditional credibility, and conditional chance.

Theorem 4.47 Let (Γ,L,M) be an uncertainty space, and B an event withMB > 0. Then M·|B defined by (4.88) is an uncertain measure, and(Γ,L,M·|B) is an uncertainty space.

Proof: It is sufficient to prove that M·|B satisfies the normality, mono-tonicity, self-duality and countable subadditivity axioms. At first, it satisfiesthe normality axiom, i.e.,

MΓ|B = 1− MΓc ∩BMB

= 1− M∅MB

= 1.



MA1 ∩BMB

≤ MA2 ∩BMB

< 0.5,

then

MA1|B =MA1 ∩BMB

≤ MA2 ∩BMB

= MA2|B.

IfMA1 ∩BMB

≤ 0.5 ≤ MA2 ∩BMB

,

then MA1|B ≤ 0.5 ≤MA2|B. If

0.5 <MA1 ∩BMB

≤ MA2 ∩BMB

,

then we have

MA1|B =(1− MAc

1 ∩BMB

)∨0.5 ≤

(1− MAc

2 ∩BMB

)∨0.5 = MA2|B.

This means that M·|B satisfies the monotonicity axiom. For any event A,if

MA ∩BMB

≥ 0.5,MAc ∩BMB

≥ 0.5,

then we have MA|B+MAc|B = 0.5 + 0.5 = 1 immediately. Otherwise,without loss of generality, suppose

MA ∩BMB

< 0.5 <MAc ∩BMB

,

then we have

MA|B+MAc|B =MA ∩BMB

+(

1− MA ∩BMB

)= 1.

That is, M·|B satisfies the self-duality axiom. Finally, for any countablesequence Ai of events, if MAi|B < 0.5 for all i, it follows from thecountable subadditivity axiom that

M

∞⋃i=1

Ai ∩B

≤M

∞⋃i=1

Ai ∩B

MB

≤

∞∑i=1

MAi ∩B

MB=

∞∑i=1

MAi|B.

Suppose there is one term greater than 0.5, say

MA1|B ≥ 0.5, MAi|B < 0.5, i = 2, 3, · · ·


If M∪iAi|B = 0.5, then we immediately have

M

∞⋃i=1

Ai ∩B

≤

∞∑i=1

MAi|B.

If M∪iAi|B > 0.5, we may prove the above inequality by the followingfacts:

Ac1 ∩B ⊂

∞⋃i=2

(Ai ∩B) ∪

( ∞⋂i=1

Aci ∩B

),

MAc1 ∩B ≤

∞∑i=2

MAi ∩B+M

∞⋂i=1

Aci ∩B

,

M

∞⋃i=1

Ai|B

= 1−

M

∞⋂i=1

Aci ∩B

MB

,

∞∑i=1

MAi|B ≥ 1− MAc1 ∩B

MB+

∞∑i=2

MAi ∩B

MB.

If there are at least two terms greater than 0.5, then the countable subad-ditivity is clearly true. Thus M·|B satisfies the countable subadditivityaxiom. Hence M·|B is an uncertain measure. Furthermore, (Γ,L,M·|B)is an uncertainty space.

Example 4.50: Let ξ and η be two uncertain variables. Then we have

M ξ = x|η = y =

Mξ = x, η = yMη = y

, ifMξ = x, η = yMη = y

< 0.5

1− Mξ 6= x, η = yMη = y

, ifMξ 6= x, η = yMη = y

< 0.5

0.5, otherwise

provided that Mη = y > 0.

Definition 4.34 (Liu [138]) Let ξ be an uncertain variable on (Γ,L,M). Aconditional uncertain variable of ξ given B is a measurable function ξ|B fromthe conditional uncertainty space (Γ,L,M·|B) to the set of real numberssuch that

ξ|B(γ) ≡ ξ(γ), ∀γ ∈ Γ. (4.90)


Definition 4.35 (Liu [138]) The conditional uncertainty distribution Φ: < →[0, 1] of an uncertain variable ξ given B is defined by

Φ(x|B) = M ξ ≤ x|B (4.91)

provided that MB > 0.

Example 4.51: Let ξ and η be uncertain variables. Then the conditionaluncertainty distribution of ξ given η = y is

Φ(x|η = y) =

Mξ ≤ x, η = yMη = y

, ifMξ ≤ x, η = yMη = y

< 0.5

1− Mξ > x, η = yMη = y

, ifMξ > x, η = yMη = y

< 0.5

0.5, otherwise

provided that Mη = y > 0.

Definition 4.36 (Liu [138]) The conditional uncertainty density function φof an uncertain variable ξ given B is a nonnegative function such that

Φ(x|B) =∫ x

−∞φ(y|B)dy, ∀x ∈ <, (4.92)

∫ +∞

−∞φ(y|B)dy = 1 (4.93)

where Φ(x|B) is the conditional uncertainty distribution of ξ given B.

Definition 4.37 (Liu [138]) Let ξ be an uncertain variable. Then the con-ditional expected value of ξ given B is defined by

E[ξ|B] =∫ +∞

0

Mξ ≥ r|Bdr −∫ 0

−∞Mξ ≤ r|Bdr (4.94)


Following conditional uncertain measure and conditional expected value,we also have conditional variance, conditional moments, conditional criticalvalues, conditional entropy as well as conditional convergence.

Chapter 5

Uncertain Process

An uncertain process is essentially a sequence of uncertain variables indexedby time or space. The study of uncertain process was started by Liu [139]in 2008. This chapter introduces the basic concepts of uncertain process,including renewal process and stationary process.

5.1 Basic Definitions

Definition 5.1 (Liu [139]) Let T be an index set and let (Γ,L,M) be anuncertainty space. An uncertain process is a measurable function from T ×(Γ,L,M) to the set of real numbers, i.e., for each t ∈ T and any Borel set Bof real numbers, the set

Xt ∈ B = γ ∈ Γ∣∣ Xt(γ) ∈ B (5.1)

is an event.

That is, an uncertain process Xt(γ) is a function of two variables suchthat the function Xt∗(γ) is an uncertain variable for each t∗.

Definition 5.2 An uncertain process Xt is said to have independent incre-ments if

Xt1 −Xt0 , Xt2 −Xt1 , · · · , Xtk−Xtk−1 (5.2)

are independent uncertain variables for any times t0 < t1 < · · · < tk. Anuncertain process Xt is said to have stationary increments if, for any givent > 0, the increments Xs+t−Xs are identically distributed uncertain variablesfor all s > 0.

Definition 5.3 For any partition of closed interval [0, t] with 0 = t1 < t2 <· · · < tk+1 = t, the mesh is written as

∆ = max1≤i≤k

|ti+1 − ti|.

202 Chapter 5 - Uncertain Process

Let m > 0 be a real number. Then the m-variation of uncertain process Xt

is

‖X‖mt = lim

∆→0

k∑i=1

|Xti−Xti

|m (5.3)

provided that the limit exists almost surely and is an uncertain process. Espe-cially, the m-variation is called total variation if m = 1; and squared variationif m = 2.

5.2 Continuity Concepts

Definition 5.4 For each fixed γ∗, the function Xt(γ∗) is called a samplepath of the uncertain process Xt.

Definition 5.5 An uncertain process Xt is said to be sample-continuous ifalmost all sample paths are continuous with respect to t.

Definition 5.6 An uncertain process Xt is said to be continuous in mean iffor every t we have

lims→t

E[|Xs −Xt|] = 0. (5.4)

Definition 5.7 An uncertain process Xt is said to be continuous in measureif for every t and ε > 0 we have

lims→t

M[|Xs −Xt| > ε] = 0. (5.5)

Definition 5.8 An uncertain process Xt is said to be continuous almostsurely if for every t we have

M

[lims→t

Xs = Xt

]= 1. (5.6)

5.3 Renewal Process

Definition 5.9 (Liu [139]) Let ξ1, ξ2, · · · be iid positive uncertain variables.Define S0 = 0 and Sn = ξ1 + ξ2 + · · · + ξn for n ≥ 1. Then the uncertainprocess

Nt = maxn≥0

n∣∣ Sn ≤ t

(5.7)

is called a renewal process.

If ξ1, ξ2, · · · denote the interarrival times of successive events. Then Sn

can be regarded as the waiting time until the occurrence of the nth event,and Nt is the number of renewals in (0, t]. Each sample path of Nt is aright-continuous and increasing step function taking only nonnegative integervalues. Furthermore, the size of each jump of Nt is always 1. In other words,

Section 5.3 - Renewal Process 203

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0 ...............

ξ1 ξ2 ξ3 ξ4

S0 S1 S2 S3 S4

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t

Nt

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Figure 5.1: A Sample Path of Renewal Process

Nt has at most one renewal at each time. In particular, Nt does not jump attime 0. Since Nt ≥ n is equivalent to Sn ≤ t, we immediately have

MNt ≥ n = MSn ≤ t. (5.8)

Theorem 5.1 Let Nt be a renewal process with uncertain interarrival times.Then we have

E[Nt] =∞∑

n=1

MSn ≤ t. (5.9)

Proof: Since Nt takes only nonnegative integer values, we have

E[Nt] =∫ ∞

0

MNt ≥ rdr =∞∑

n=1

∫ n

n−1

MNt ≥ rdr

=∞∑

n=1

MNt ≥ n =∞∑

n=1

MSn ≤ t.


Example 5.1: A renewal process Nt is called an rectangular renewal processif ξ1, ξ2, · · · are iid rectangular uncertain variables (a, b) with a > 0. Thenfor each nonnegative integer n, we have

MNt ≥ n =

0, if t < na

0.5, if na ≤ t < nb

1, if t ≥ nb,

(5.10)

E[Nt] =12

(⌊t

a

⌋+⌊t

b

⌋)(5.11)

where bxc represents the maximal integer less than or equal to x.

204 Chapter 5 - Uncertain Process

Example 5.2: A renewal process Nt is called a triangular renewal processif ξ1, ξ2, · · · are iid triangular uncertain variables (a, b, c) with a > 0. Thenfor each nonnegative integer n, we have

MNt ≥ n =

0, if t ≤ na

t− na

2n(b− a), if na ≤ t ≤ nb

nc− 2nb+ t

2n(c− b), if nb ≤ t ≤ nc

1, if t ≥ nc.

(5.12)

Theorem 5.2 (Renewal Theorem) Let Nt be a renewal process with uncer-tain interarrival times ξ1, ξ2, · · · Then

limt→∞

E[Nt]t

= E

[1ξ1

]. (5.13)

Proof: Since ξ1 is a positive uncertain variable and Nt is a nonnegativeuncertain variable, we have

E[Nt]t

=∫ ∞

0

M

Nt

t≥ r

dr, E

[1ξ1

]=∫ ∞

0

M

1ξ1≥ r

dr.

It is easy to verify that

M

Nt

t≥ r

≤M

1ξ1≥ r

and

limt→∞

M

Nt

t≥ r

= M

1ξ1≥ r

for any real numbers t > 0 and r > 0. It follows from Lebesgue dominatedconvergence theorem that

limt→∞

∫ ∞

0

M

Nt

t≥ r

dr =

∫ ∞

0

M

1ξ1≥ r

dr.

Hence (5.13) holds.

5.4 Stationary Process

Definition 5.10 An uncertain process Xt is called stationary if for any pos-itive integer k and any times t1, t2, · · · , tk and s, the uncertain vectors

(Xt1 , Xt2 , · · · , Xtk) and (Xt1+s, Xt2+s, · · · , Xtk+s) (5.14)

are identically distributed.

Chapter 6

Uncertain Calculus

Uncertain calculus, founded by Liu [139] in 2008 and revised by Liu [141] in2009, is composed of canonical process, uncertain integral and chain rule.

6.1 Canonical Process

Definition 6.1 (Liu [139][141]) An uncertain process Ct is said to be acanonical process if(i) C0 = 0 and Ct is sample-continuous,(ii) Ct has stationary and independent increments,(iii) every increment Cs+t −Cs is a normal uncertain variable with expectedvalue 0 and variance t2.

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t

Ct

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Figure 6.1: A Sample Path of Canonical Process

Remark 6.1: In 1828 the botanist Brown observed irregular movement ofpollen suspended in liquid. A rigorous mathematical definition of Brownianmotion was given by Wiener in 1931. This movement is now known as Brow-nian motion. Please mention that Brownian motion is not a special canonicalprocess.

206 Chapter 6 - Uncertain Calculus

Theorem 6.1 (Existence Theorem) There is a canonical process.

Proof: Without loss of generality, we only prove that there is a canonicalprocess on the range of t ∈ [0, 1]. Let

ξ(r)∣∣ r represents rational numbers in [0, 1]

be a countable sequence of independently normal uncertain variables withexpected value zero and variance one. For each integer n, we define anuncertain process

Xn(t) =

1n

k∑i=1

ξ

(i

n

), if t =

k

n(k = 0, 1, · · · , n)

linear, otherwise.

Since the limitlim

n→∞Xn(t)

exists almost surely, we may verify that the limit meets the conditions ofcanonical process. Hence there is a canonical process.

Let Ct be a canonical process, and dt an infinitesimal time interval. ThendCt = Ct+dt − Ct is an uncertain process with E[dCt] = 0 and

dt2

2≤ E[dC2

t ] ≤ dt2. (6.1)

Theorem 6.2 (Reflection Principle) Let Ct be a canonical process. Thenfor any level x > 0 and any time t > 0, we have

MCt ≥ x ≤M

max0≤s≤t

Cs ≥ x

≤ 2MCt ≥ x. (6.2)

For any level x < 0 and any time t > 0, we have

MCt ≤ x ≤M

max0≤s≤t

Cs ≤ x

≤ 2MCt ≤ x. (6.3)

Arithmetic Canonical Process

Definition 6.2 Let Ct be a canonical process. Then for any real numbers eand σ,

At = et+ σCt (6.4)

is called an arithmetic canonical process, where e is called the drift and σ iscalled the diffusion.

It is easy to verify that the expected value E[At] = et and varianceV [At] = σ2t2 at any time t.

Section 6.2 - Uncertain Integral 207

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Ct

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Figure 6.2: Reflection Principle of Canonical Process

Geometric Canonical Process

Definition 6.3 Let Ct be a canonical process. Then et+σCt is an arithmeticcanonical process, and the uncertain process

Gt = exp(et+ σCt) (6.5)

is called a geometric canonical process, where e is called the log-drift and σis called the log-diffusion.

6.2 Uncertain Integral

Definition 6.4 (Liu [141]) Let Xt be an uncertain process and let Ct be acanonical process. For any partition of closed interval [a, b] with a = t1 <t2 < · · · < tk+1 = b, the mesh is written as

∆ = max1≤i≤k

|ti+1 − ti|.

Then the uncertain integral of Xt with respect to Ct is∫ b

a

XtdCt = lim∆→0

k∑i=1

Xti· (Cti+1 − Cti

) (6.6)

provided that the limit exists almost surely and is an uncertain variable.

Example 6.1: Let Ct be a canonical process. Then for any partition 0 =t1 < t2 < · · · < tk+1 = s, we have∫ s

0

dCt = lim∆→0

k∑i=1

(Cti+1 − Cti) ≡ Cs − C0 = Cs.


Example 6.2: Let Ct be a canonical process. Then for any partition 0 =t1 < t2 < · · · < tk+1 = s, we have

sCs =k∑

i=1

(ti+1Cti+1 − tiCti

)=

k∑i=1

ti(Cti+1 − Cti) +k∑

i=1

Cti+1(ti+1 − ti)

→∫ s

0

tdCt +∫ s

0

Ctdt

as ∆ → 0. It follows that∫ s

0

tdCt = sCs −∫ s

0

Ctdt.

Example 6.3: Let Ct be a canonical process. Then for any partition 0 =t1 < t2 < · · · < tk+1 = s, we have

C2s =

k∑i=1

(C2

ti+1− C2

ti

)

=k∑

i=1

(Cti+1 − Cti

)2 + 2k∑

i=1

Cti

(Cti+1 − Cti

)→ 0 + 2

∫ s

0

CtdCt

as ∆ → 0. That is, ∫ s

0

CtdCt =12C2

s .

Example 6.4: Let Ct be a canonical process. Then for any number α(0 < α < 1), the uncertain process

Ft =∫ t

0

(t− s)αdCs (6.7)

is called a fractional canonical process with index α.

6.3 Chain Rule

Theorem 6.3 (Liu [141]) Let Ct be a canonical process, and let h(t, c) be acontinuously differentiable function. Define Xt = h(t, Ct). Then we have thefollowing chain rule

dXt =∂h

∂t(t, Ct)dt+

∂h

∂c(t, Ct)dCt. (6.8)

Section 6.3 - Chain Rule 209

Proof: Since the function h is continuously differentiable, by using Taylorseries expansion, the infinitesimal increment of Xt has a first-order approxi-mation

∆Xt =∂h

∂t(t, Ct)∆t+

∂h

∂c(t, Ct)∆Ct.

Hence we obtain the chain rule because it makes

Xs = X0 +∫ s

0

∂h

∂t(t, Ct)dt+

∫ s

0

∂h

∂c(t, Ct)dCt

for any s ≥ 0.

Remark 6.2: The infinitesimal increment dCt in (6.8) may be replaced withthe derived canonical process

dYt = utdt+ vtdCt (6.9)

where ut and vt are absolutely integrable uncertain processes, thus producing

dh(t, Yt) =∂h

∂t(t, Yt)dt+

∂h

∂c(t, Yt)dYt. (6.10)

Example 6.5: Applying the chain rule, we obtain the following formula

d(tCt) = Ctdt+ tdCt.

Hence we have

sCs =∫ s

0

d(tCt) =∫ s

0

Ctdt+∫ s

0

tdCt.

That is, ∫ s

0

tdCt = sCs −∫ s

0

Ctdt.


d(C2t ) = 2CtdCt.

Then we haveC2

s =∫ s

0

d(C2t ) = 2

∫ s

0

CtdCt.

It follows that ∫ s

0

CtdCt =12C2

s .


d(C3t ) = 3C2

t dCt.


Thus we get

C3s =

∫ s

0

d(C3t ) = 3

∫ s

0

C2t dCt.

That is ∫ s

0

C2t dCt =

13C3

s .

6.4 Integration by Parts

Theorem 6.4 (Liu [139], Integration by Parts) Suppose that Ct is a canon-ical process and F (t) is an absolutely continuous function. Then∫ s

0

F (t)dCt = F (s)Cs −∫ s

0

CtdF (t). (6.11)

Proof: By defining h(t, Ct) = F (t)Ct and using the chain rule, we get

d(F (t)Ct) = CtdF (t) + F (t)dCt.

ThusF (s)Cs =

∫ s

0

d(F (t)Ct) =∫ s

0

CtdF (t) +∫ s

0

F (t)dCt

which is just (6.11).

Example 6.8: Assume F (t) ≡ 1. Then by using the integration by parts,we immediately obtain ∫ s

0

dCt = Cs.

Example 6.9: Assume F (t) = t. Then by using the integration by parts,we immediately obtain ∫ s

0

tdCt = sCs −∫ s

0

Ctdt.

Example 6.10: Assume F (t) = t2. Then by using the integration by parts,we obtain ∫ s

0

t2dCt = s2Cs −∫ s

0

Ctdt2 = s2Cs − 2∫ s

0

tCtdt

= (s2 − 2s)Cs + 2∫ s

0

Ctdt.

Example 6.11: Assume F (t) = sin t. Then by using the integration byparts, we obtain∫ s

0

sin tdCt = sin s · Cs −∫ s

0

Ctd sin t = sin s · Cs −∫ s

0

cos t · Ctdt.

Chapter 7

Uncertain DifferentialEquation

Uncertain differential equation, proposed by Liu [139] in 2008, is a type ofdifferential equation driven by canonical process. This chapter will discussthe existence, uniqueness and stability of solutions of uncertain differentialequations. This chapter also presents some applications of uncertain differen-tial equation in uncertain finance, uncertain control, uncertain filtering, anduncertain dynamical system.

7.1 Uncertain Differential Equation

Definition 7.1 (Liu [139]) Suppose Ct is a canonical process, and f and gare some given functions. Then

dXt = f(t,Xt)dt+ g(t,Xt)dCt (7.1)

is called an uncertain differential equation. A solution is an uncertain processXt that satisfies (7.1) identically in t.

Remark 7.1: Note that there is no precise definition for the terms dXt,dt and dCt in the uncertain differential equation (7.1). The mathematicallymeaningful form is the uncertain integral equation

Xs = X0 +∫ s

0

f(t,Xt)dt+∫ s

0

g(t,Xt)dCt. (7.2)

However, the differential form is more convenient for us. This is the mainreason why we accept the differential form.

212 Chapter 7 - Uncertain Differential Equation

Example 7.1: Let Ct be a canonical process. Then the uncertain differentialequation

dXt = adt+ bdCt

has a solution Xt = at+ bCt which is just an arithmetic canonical process.


dXt = aXtdt+ bXtdCt

has a solutionXt = exp (at+ bCt)

which is just a geometric canonical process.


dXt = (aXt + b)dt+ cdCt

has a solution

Xt = exp(at)(X0 +

b

a(1− exp(−at)) + c

∫ t

0

exp(−as)dCs

)provided that a 6= 0.

7.2 Existence and Uniqueness Theorem

Theorem 7.1 (Existence and Uniqueness Theorem) The uncertain differen-tial equation (7.1) has a unique solution if the coefficients f(x, t) and g(x, t)satisfy the Lipschitz condition

|f(x, t)− f(y, t)|+ |g(x, t)− g(y, t)| ≤ L|x− y|, ∀x, y ∈ <, t ≥ 0 (7.3)

and linear growth condition

|f(x, t)|+ |g(x, t)| ≤ L(1 + |x|), ∀x ∈ <, t ≥ 0 (7.4)

for some constant L. Moreover, the solution is sample-continuous.

7.3 Stability

Definition 7.2 (Liu [141]) An uncertain differential equation is said to bestable if for any given ε > 0 and ε > 0, there exists δ > 0 such that for anysolutions Xt and Yt, we have

M|Xt − Yt| > ε < ε, ∀t > 0 (7.5)

whenever |X0 − Y0| < δ.

Section 7.4 - Uncertain Finance 213

Example 7.4: The uncertain differential equation dXt = adt+bdCt is stablesince for any given ε > 0 and ε > 0, we may take δ = ε and have

M|Xt − Yt| > ε = M|X0 − Y0| > ε = M∅ = 0 < ε

for any t > 0 whenever |X0 − Y0| < δ.

Example 7.5: The uncertain differential equation dXt = Xtdt + bdCt isunstable since for any given ε > 0 and any different initial solutions X0 andY0, we have

M|Xt − Yt| > ε = Mexp(t)|X0 − Y0| > ε = 1

provided that t is sufficiently large.

7.4 Uncertain Finance

If we assume that the stock price follows some uncertain differential equation,then we may produce a new topic of uncertain finance. As an example, letstock price follow geometric canonical process. Then we have a stock model(Liu [141]) in which the bond price Xt and the stock price Yt are determinedby

dXt = rXtdt

dYt = eYtdt+ σYtdCt

(7.6)

where r is the riskless interest rate, e is the stock drift, σ is the stock diffusion,and Ct is a canonical process.

European Call Option Price

A European call option gives the holder the right to buy a stock at a speci-fied time for specified price. Assume that the option has strike price K andexpiration time s. Then the payoff from such an option is (Ys −K)+. Con-sidering the time value of money resulted from the bond, the present valueof this payoff is exp(−rs)(Ys −K)+. Hence the European call option priceshould be

fc = exp(−rs)E[(Ys −K)+]. (7.7)

It is clear that the option price is a decreasing function of interest rate r.That is, the European call option will devaluate if the interest rate is raised;and the European call option will appreciate in value if the interest rate isreduced. In addition, the option price is also a decreasing function of strikeprice K.


Let us consider the financial market described by stock model (7.6). TheEuropean call option price is

fc = exp(−rs)E[(Y0 exp(es+ σCs)−K)+]

= exp(−rs)∫ +∞

0

MY0 exp(es+ σCs)−K ≥ xdx

= exp(−rs)Y0

∫ +∞

K/Y0

Mexp(es+ σCs) ≥ ydy

= exp(−rs)Y0

∫ +∞

K/Y0

Mes+ σCs ≥ ln ydy

= exp(−rs)Y0

∫ +∞

K/Y0

(1 + exp

(π(ln y − es)√

3σs

))−1

dy

which produces a European call option price formula (Liu [141])

fc = exp(−rs)Y0

∫ +∞

K/Y0

(1 + exp

(π(ln y − es)√

3σs

))−1

dy. (7.8)

European Put Option Price

A European put option gives the holder the right to sell a stock at a speci-fied time for specified price. Assume that the option has strike price K andexpiration time s. Then the payoff from such an option is (K − Ys)+. Con-sidering the time value of money resulted from the bond, the present valueof this payoff is exp(−rs)(K − Ys)+. Hence the European put option priceshould be

fp = exp(−rs)E[(K − Ys)+]. (7.9)

It is easy to verify that the option price is a decreasing function of interestrate r, and is an increasing function of strike price K.

Let us consider the financial market described by stock model (7.6). TheEuropean put option price is

fp = exp(−rs)E[(K − Y0 exp(es+ σCs))+]

= exp(−rs)∫ +∞

0

MK − Y0 exp(es+ σCs) ≥ xdx

= exp(−rs)Y0

∫ +∞

K/Y0

Mexp(es+ σCs) ≤ ydy

= exp(−rs)Y0

∫ K/Y0

0

Mes+ σCs ≤ ln ydy

= exp(−rs)Y0

∫ K/Y0

0

(1 + exp

(π(es− ln y)√

3σs

))−1

dy

Section 7.4 - Uncertain Finance 215

which produces a European put option price formula

fp = exp(−rs)Y0

∫ K/Y0

0

(1 + exp

(π(es− ln y)√

3σs

))−1

dy. (7.10)

Multi-factor Stock Model

Now we assume that there are multiple stocks whose prices are determinedby multiple canonical processes. For this case, we have a multi-factor stockmodel (Liu [141]) in which the bond price Xt and the stock prices Yit aredetermined by

dXt = rXtdt

dYit = eiYitdt+n∑

j=1

σijYitdCjt, i = 1, 2, · · · ,m(7.11)

where r is the riskless interest rate, ei are the stock drift coefficients, σij

are the stock diffusion coefficients, Cit are independent canonical processes,i = 1, 2, · · · ,m, j = 1, 2, · · · , n.

Portfolio Selection

For the stock model (7.11), we have the choice of m+1 different investments.At each instant t we may choose a portfolio (βt, β1t, · · · , βmt) (i.e., the in-vestment fractions meeting βt + β1t + · · ·+ βmt = 1). Then the wealth Zt attime t should follow uncertain differential equation

dZt = rβtZtdt+m∑

i=1

eiβitZtdt+m∑

i=1

n∑j=1

σijβitZtdCjt. (7.12)

Portfolio selection problem is to find an optimal portfolio (βt, β1t, · · · , βmt)such that the expected wealth E[Zs] is maximized and variance V [Zs] isminimized at terminal time s. In order to balance the two objectives, we may,for example, maximize the expected wealth subject to a variance constraint,i.e.,

maxE[Zs]subject to:

V [Zs] ≤ V

(7.13)

where V is a given level. This is just the so-called mean-variance model infinance.


No-Arbitrage

The stock model (7.11) is said to be no-arbitrage if there is no portfolio(βt, β1t, · · · , βmt) such that for some time s > 0, we have

Mexp(−rs)Zs ≥ Z0 = 1 (7.14)

andMexp(−rs)Zs > Z0 > 0 (7.15)

where Zt is determined by (7.12) and represents the wealth at time t. We mayprove that the stock model (7.11) is no-arbitrage if and only if its diffusionmatrix

σ11 σ12 · · · σ1n

σ21 σ22 · · · σ2n

......

. . ....

σm1 σm2 · · · σmn

has rank m, i.e., the row vectors are linearly independent.

7.5 Uncertain Control

A control system is assumed to follow the uncertain differential equation

dXt = f(t,Xt, Zt)dt+ g(t,Xt, Zt)dCt (7.16)

where Xt is the state and Zt is a control. An uncertain control theory isneeded to deal with this type of uncertain control system.

Assume that R is the return function and T is the function of terminalreward. If we want to maximize the expected return on [0, s] by using anoptimal control, then we have the following control model,

maxZt

E

[∫ s

0

R(t,Xt, Zt)dt+ T (s,Xs)]

subject to:dXt = f(t,Xt, Zt)dt+ g(t,Xt, Zt)dCt.

(7.17)

Hamilton-Jacobi-Bellman equation provides a necessary condition for ex-tremum of stochastic control model, and Zhu’s equation [279] provides anecessary condition for extremum of fuzzy control model. What is the nec-essary condition for extremum of general control model? How do we find theoptimal control?

7.6 Uncertain Filtering

Suppose an uncertain system Xt is described by an uncertain differentialequation


Section 7.7 - Uncertain Dynamical System 217

where Ct is a canonical process. We also have an observation Xt with


where Ct is another canonical process that is independent of Ct. The uncer-tain filtering problem is to find the best estimate of Xt based on the obser-vation Xt.

One outstanding contribution to stochastic filtering problem is Kalman-Bucy filter. How do we filter the noise away from the observation for generaluncertain process?

7.7 Uncertain Dynamical System

Usually a stochastic dynamical system is described by a stochastic differentialequation, and a fuzzy dynamical system is described by a fuzzy differentialequation. Here we define an uncertain dynamical system as an uncertaindifferential equation. Especially, a first-order uncertain system is a first-orderuncertain differential equation

Xt = f(t,Xt) + g(t,Xt)Ct, (7.20)

and a second-order uncertain system is a second-order uncertain differentialequation

Xt = f(t,Xt, Xt) + g(t,Xt, Xt)Ct, (7.21)

where

Xt =dXt

dt, Xt =

dXt

dt, Ct =

dCt

dt. (7.22)

We should develop a new theory for uncertain dynamical systems.

Chapter 8

Uncertain Logic

Uncertain logic was designed by Li and Liu [108] in 2009 as a generalizationof logic for dealing with uncertain knowledge. It is essentially the uncertaintytheory in which uncertain variables take values either 0 or 1. A key point inuncertain logic is that the truth value of an uncertain proposition is definedas the uncertain measure that the proposition is true. One advantage ofuncertain logic is the well consistency with classical logic!

8.1 Uncertain Proposition

Definition 8.1 An uncertain proposition is a statement with truth value be-longing to [0, 1].

Example 8.1: “Tom is tall with truth value 0.8” is an uncertain proposition,where “Tom is tall” is a statement, and its truth value is 0.8 in uncertainmeasure.

Generally, we use ξ to express the uncertain proposition and use c toexpress its uncertain measure value. In fact, the uncertain proposition ξ isessentially an uncertain variable

ξ =

1 with uncertain measure c0 with uncertain measure 1− c

(8.1)

where ξ = 1 means ξ is true and ξ = 0 means ξ is false.

Definition 8.2 Uncertain propositions are called independent if they are in-dependent uncertain variables.

220 Chapter 8 - Uncertain Logic

Example 8.2: Two uncertain propositions ξ and η are independent if andonly if they are independent uncertain variables, i.e.,

M(ξ = 1) ∩ (η = 1) = Mξ = 1 ∧Mη = 1,M(ξ = 1) ∩ (η = 0) = Mξ = 1 ∧Mη = 0,M(ξ = 0) ∩ (η = 1) = Mξ = 0 ∧Mη = 1,M(ξ = 0) ∩ (η = 0) = Mξ = 0 ∧Mη = 0.

Or equivalently,

M(ξ = 1) ∪ (η = 1) = Mξ = 1 ∨Mη = 1,M(ξ = 1) ∪ (η = 0) = Mξ = 1 ∨Mη = 0,M(ξ = 0) ∪ (η = 1) = Mξ = 0 ∨Mη = 1,M(ξ = 0) ∪ (η = 0) = Mξ = 0 ∨Mη = 0.

8.2 Uncertain Formula

An uncertain formula is a finite sequence of uncertain propositions and con-nective symbols that must make sense. For example, let ξ, η, τ be uncertainpropositions. Then

X = ¬ξ, X = ξ ∧ η, X = (ξ ∨ η) → τ

are all uncertain formulas. Note that an uncertain formula X is essentiallyan uncertain variable taking values 0 or 1. If X = 1, then X is true; if X = 0,then X is false.

Definition 8.3 Uncertain formulas are called independent if they are inde-pendent uncertain variables.

8.3 Truth Value

Truth value is a key concept in uncertain logic, and is defined as the uncertainmeasure that the uncertain formula is true.

Definition 8.4 (Li and Liu [108]) Let X be an uncertain formula. Thenthe truth value of X is defined as the uncertain measure that the uncertainformula X is true, i.e.,

T (X) = MX = 1. (8.2)

The truth value T (X) = 1 means X is certainly true, T (X) = 0 means Xis certainly false, and T (X) = 0.5 means X is totally uncertain. The higherthe truth value is, the more true the uncertain formula is.

Section 8.4 - Some Laws 221

Example 8.3: Let ξ and η be two independent uncertain propositions withuncertain measure values a and b, respectively. Then

T (ξ) = Mξ = 1 = a, (8.3)

T (¬ξ) = Mξ = 0 = 1− a, (8.4)

T (ξ ∨ η) = Mξ ∨ η = 1 = M(ξ = 1) ∪ (η = 1) = a ∨ b, (8.5)

T (ξ ∧ η) = Mξ ∧ η = 1 = M(ξ = 1) ∩ (η = 1) = a ∧ b, (8.6)

T (ξ → η) = T (¬ξ ∨ η) = (1− a) ∨ b. (8.7)

Theorem 8.1 (Chen [21]) Let X be an uncertain formula containing inde-pendent uncertain propositions ξ1, ξ2, · · · , ξn whose truth function is f . Thenthe truth value of X is

T (X) =

supf(x1,x2,··· ,xn)=1

min1≤i≤n

νi(xi),

if supf(x1,x2,··· ,xn)=1

min1≤i≤n

νi(xi) < 0.5

1− supf(x1,x2,··· ,xn)=0

min1≤i≤n

νi(xi),


min1≤i≤n

νi(xi) ≥ 0.5

(8.8)

where νi(xi) are identification functions (no matter what types they are) ofξi, i = 1, 2, · · · , n, respectively.

Proof: Since X = 1 if and only if f(ξ1, ξ2, · · · , ξn) = 1, the formula of truthvalue follows from the operational law immediately.

8.4 Some Laws

Theorem 8.2 (Law of Contradiction) For any uncertain formula X, wehave

T (X ∧ ¬X) = 0. (8.9)

Proof: It follows from the definition of truth value and property of uncertainmeasure that

T (X ∧ ¬X) = MX ∧ ¬X = 1 = M(X = 1) ∩ (X = 0) = M∅ = 0.


Theorem 8.3 (Law of Excluded Middle) For any uncertain formula X, wehave

T (X ∨ ¬X) = 1. (8.10)


Proof: It follows from the definition of truth value and property of uncertainmeasure that

T (X ∨ ¬X) = MX ∨ ¬X = 1 = M(X = 1) ∪ (X = 0) = 1.


Theorem 8.4 (Law of Truth Conservation) For any uncertain formula X,we have

T (X) + T (¬X) = 1. (8.11)

Proof: It follows from the self-duality of uncertain measure that

T (¬X) = M¬X = 1 = MX = 0 = 1−MX = 1 = 1− T (X).


Theorem 8.5 (De Morgan’s Law) For any uncertain formulas X and Y , wehave

T (¬(X ∧ Y )) = T ((¬X) ∨ (¬Y )), (8.12)

T (¬(X ∨ Y )) = T ((¬X) ∧ (¬Y )). (8.13)

Proof: It follows from the basic properties of uncertain measure that

T (¬(X ∧ Y )) = MX ∧ Y = 0 = M(X = 0) ∩ (Y = 0)= M(¬X) ∨ (¬Y ) = 1 = T ((¬X) ∨ (¬Y ))

which proves the first equality. A similar way may verify the second equality.The theorem is proved.

Theorem 8.6 (Law of Contraposition) For any uncertain formulas X andY , we have

T (X → Y ) = T (¬Y → ¬X). (8.14)

Proof: It follows from the basic properties of uncertain measure that

T (X → Y ) = M(¬X) ∨ Y = 1 = M(X = 0) ∪ (Y = 1)= MY ∨ (¬X) = 1 = T (¬Y → ¬X).


Theorem 8.7 (Monotonicity and Subadditivity) For any uncertain formulasX and Y , we have

T (X) ∨ T (Y ) ≤ T (X ∨ Y ) ≤ T (X) + T (Y ). (8.15)

Section 8.4 - Some Laws 223

Proof: It follows from the monotonicity of uncertain measure that

T (X ∨ Y ) = MX ∨ Y = 1 = M(X = 1) ∪ (Y = 1)

≥MX = 1 ∨MY = 1 = T (X) ∨ T (Y ).

It follows from the subadditivity of uncertain measure that

T (X ∨ Y ) = MX ∨ Y = 1 = M(X = 1) ∪ (Y = 1)

≤MX = 1+MY = 1 = T (X) + T (Y ).

The theorem is verified.

Theorem 8.8 For any uncertain formulas X and Y , we have

T (X) + T (Y )− 1 ≤ T (X ∧ Y ) ≤ T (X) ∧ T (Y ). (8.16)

Proof: It follows from the monotonicity of truth value that

T (X ∧ Y ) = 1− T (¬X ∨ ¬Y ) ≤ 1− T (¬X) ∨ T (¬Y )= (1− T (¬X)) ∧ (1− T (¬Y )) = T (X) ∧ T (Y ).

It follows from the subadditivity of truth value that

T (X ∧ Y ) = 1− T (¬X ∨ ¬Y ) ≥ 1− (T (¬X) + T (¬Y ))= 1− (1− T (X))− (1− T (Y )) = T (X) + T (Y )− 1.


Theorem 8.9 For any uncertain formula X, we have

T (X → X) = 1, T (X → ¬X) = 1− T (X). (8.17)

Proof: It follows from the law of excluded middle and law of truth conser-vation that

T (X → X) = T (¬X ∨X) = 1,

T (X → ¬X) = T (¬X ∨ ¬X) = T (¬X) = 1− T (X).


Independence Case

Theorem 8.10 If uncertain formulas X and Y are independent, then wehave

T (X ∨ Y ) = T (X) ∨ T (Y ), T (X ∧ Y ) = T (X) ∧ T (Y ). (8.18)


Proof: Since X and Y are independent uncertain formulas, they are inde-pendent uncertain variables. Hence

T (X ∨ Y ) = MX ∨ Y = 1 = MX = 1 ∨MY = 1 = T (X) ∨ T (Y ),T (X ∧ Y ) = MX ∧ Y = 1 = MX = 1 ∧MY = 1 = T (X) ∧ T (Y ).


Theorem 8.11 If uncertain formulas X and Y are independent, then wehave

T (X → Y ) = (1− T (X)) ∨ T (Y ). (8.19)

Proof: Since X and Y are independent, the ¬X and Y are also independent.It follows that

T (X → Y ) = T (¬X ∨ Y ) = T (¬X) ∨ T (Y ) = (1− T (X)) ∨ T (Y )

which proves the theorem.

Chapter 9

Uncertain Inference

Uncertain inference was proposed by Liu [141] in 2009. How do we inferY from the uncertain knowledge X and X → Y? This is the main issue ofuncertain inference. A key point is that the inference rules use the tool ofconditional uncertainty.

9.1 Uncertain Sets

Before introducing uncertain inference, let us recall the concept of uncertainset proposed by Liu [138] in 2007.

Definition 9.1 (Liu [138]) An uncertain set is a measurable function ξ froman uncertainty space (Γ,L,M) to a collection of sets of real numbers, i.e., forany Borel set B, the set

γ ∈ Γ∣∣ ξ(γ) ∩B 6= ∅ (9.1)

is an event.

Let ξ and ξ∗ be two uncertain sets. In order to define the degree that ξ∗

matches ξ, we first introduce the following symbols:

ξ∗ ⊂ ξ = γ ∈ Γ∣∣ ξ∗(γ) ⊂ ξ(γ), (9.2)

ξ∗ ⊂ ξc = γ ∈ Γ∣∣ ξ∗(γ) ⊂ ξ(γ)c, (9.3)

ξ∗ 6⊂ ξc = γ ∈ Γ∣∣ ξ∗(γ) ∩ ξ(γ) 6= ∅. (9.4)

What are concerned is to know the degree that ξ∗ belongs to ξ. Note thatwe cannot use the symbol “⊂” because it has certain meaning in classical settheory and is not applicable to uncertain sets. Thus we need a new symbol“ξ∗ B ξ” to represent the belongingness that reads as “ξ∗ matches ξ”.

226 Chapter 9 - Uncertain Inference

Definition 9.2 Let ξ and ξ∗ be two uncertain sets. Then the event ξ∗ B ξ(i.e., ξ∗ matches ξ) is defined as

ξ∗ B ξ =

ξ∗ ⊂ ξ, if Mξ∗ ⊂ ξ >Mξ∗ ⊂ ξc

ξ∗ 6⊂ ξc, if Mξ∗ ⊂ ξ ≤Mξ∗ ⊂ ξc.(9.5)

That is, we take the event of ξ∗ ⊂ ξ or ξ∗ 6⊂ ξc such that its uncertainmeasure is as closed to 0.5 as possible. This idea coincides with the maximumuncertainty principle.

Definition 9.3 Let ξ and ξ∗ be two uncertain sets. Then the matching degreeof ξ∗ B ξ is defined as

Mξ∗ B ξ =

Mξ∗ ⊂ ξ, if Mξ∗ ⊂ ξ >Mξ∗ ⊂ ξc

Mξ∗ 6⊂ ξc, if Mξ∗ ⊂ ξ ≤Mξ∗ ⊂ ξc.(9.6)

9.2 Inference Rule

Let X and Y be two concepts. It is assumed that the perfect relation betweenX and Y is not known. Here we suppose that we only have a rule “if X is ξthen Y is η” where ξ and η are two uncertain sets (not uncertain variable!).From X = ξ∗ we infer that Y is

η∗ = η|ξ∗Bξ (9.7)

which is the conditional uncertain set of η given ξ∗ B ξ. Thus we have thefollowing inference rule (Liu [141]):

Rule: If X is ξ then Y is ηFrom: X is ξ∗

Infer: Y is η∗ = η|ξ∗Bξ

(9.8)

Theorem 9.1 Assume the if-then rule “if X is ξ then Y is η”. From Mξ∗Bξ = 1 we infer η∗ = η.

Proof: It follows from Mξ∗ B ξ = 1 and the definition of conditionaluncertainty that η∗ = η.

Example 9.1: Suppose ξ and ξ∗ are independent uncertain sets with firstidentification functions λ and λ∗, and η has a first identification function νthat is independent of (ξ, ξ∗). If λ∗(x) ≤ λ(x) for any x ∈ <, thenMξ∗Bξ =1 and ν∗(y) = ν(y). This means that if ξ∗ matches ξ, then η∗ matches η.

Example 9.2: If the supports of ξ and ξ∗ are disjoint, then Mξ∗ B ξ = 0.For this case, we set ν∗(y) ≡ 0.5. This means that “if nothing is known thenanything is possible”.

Section 9.5 - General Inference Rule 227

9.3 Inference Rule with Multiple Antecedents

Assume that we have a rule “if X is ξ and Y is η then Z is τ”. From X = ξ∗

and Y = η∗ we infer that Z is

τ∗ = τ |(ξ∗Bξ)∩(η∗Bη) (9.9)

which is the conditional uncertain set of τ given ξ∗ B ξ and η∗ B η. Thus wehave the following inference rule:

Rule: If X is ξ and Y is η then Z is τFrom: X is ξ∗ and Y is η∗

Infer: Z is τ∗ = τ |(ξ∗Bξ)∩(η∗Bη)

(9.10)

Theorem 9.2 Assume the if-then rule “if X is ξ and Y is η then Z is τ”.From Mξ∗ B ξ = 1 and Mη∗ B η = 1 we infer τ∗ = τ .

Proof: IfMξ∗Bξ = 1 andMη∗Bη = 1, thenM(ξ∗Bξ)∩(η∗Bη) = 1.It follows from the definition of conditional uncertainty that τ∗ = τ .

9.4 Inference Rule with Multiple If-Then Rules

Assume that we have two rules “if X is ξ1 then Y is η1” and “if X is ξ2 thenY is η2”. From the rule “if X is ξ1 then Y is η1” and X = ξ∗ we infer that Yis

η∗1 = η1|ξ∗1Bξ.

From the rule “if X is ξ2 then Y is η2” and X = ξ∗ we infer that Y is

η∗2 = η2|ξ∗2Bξ.

The uncertain sets η∗1 and η∗2 will aggregate a new η∗ that is determined by

Mη∗ ⊂ B =

Mη∗1 ⊂ B ∧Mη∗2 ⊂ B,

if Mη∗1 ⊂ B+Mη∗2 ⊂ B < 1

Mη∗1 ⊂ B ∨Mη∗2 ⊂ B,

if Mη∗1 ⊂ B+Mη∗2 ⊂ B ≥ 1.

(9.11)

for any Borel set B. That is, we assign it the value of Mη∗1 ⊂ B orMη∗2 ⊂ B such that it is as far from 0.5 as possible. Thus we infer Y = η∗.Hence we have the following inference rule:

Rule 1: If X is ξ1 then Y is η1Rule 2: If X is ξ2 then Y is η2From: X is ξ∗

Infer: Y is η∗ determined by (9.11)

(9.12)

228 Chapter 9 - Uncertain Inference

9.5 General Inference Rule

As an exercise, please the reader gives an inference rule for the followingproblem:

Rule 1: If X1 is ξ11 and X2 is ξ12 and · · · and Xm is ξ1m then Y is η1Rule 2: If X1 is ξ21 and X2 is ξ22 and · · · and Xm is ξ2m then Y is η2

· · ·Rule n: If X1 is ξn1 and X2 is ξn2 and · · · and Xm is ξnm then Y is ηn

From: X1 is ξ∗1 and X2 is ξ∗2 and · · · and Xm is ξ∗mInfer: Y is η∗

What is the uncertain set η∗?

Appendix A

Measure and Integral

A.1 Measurable Sets

Algebra and σ-algebra are very important and fundamental concepts in mea-sure theory.

Definition A.1 Let Ω be a nonempty set. A collection A is called an algebraover Ω if the following conditions hold:(a) Ω ∈ A;(b) if A ∈ A, then Ac ∈ A;(c) if Ai ∈ A for i = 1, 2, · · · , n, then ∪n

i=1Ai ∈ A.If the condition (c) is replaced with closure under countable union, then A iscalled a σ-algebra over Ω.

Example A.1: Assume that Ω is a nonempty set. Then ∅,Ω is thesmallest σ-algebra over Ω, and the power set P (all subsets of Ω) is thelargest σ-algebra over Ω.

Example A.2: Let A be the set of all finite disjoint unions of all intervalsof the form (−∞, a], (a, b], (b,∞) and ∅. Then A is an algebra over <, butnot a σ-algebra because Ai = (0, (i− 1)/i] ∈ A for all i but

∞⋃i=1

Ai = (0, 1) 6∈ A.

Theorem A.1 A σ-algebra A is closed under difference, countable union,countable intersection, upper limit, lower limit, and limit. That is,

A2 \A1 ∈ A;∞⋃

i=1

Ai ∈ A;∞⋂

i=1

Ai ∈ A; (A.1)

lim supi→∞

Ai =∞⋂

k=1

∞⋃i=k

Ai ∈ A; (A.2)

230 Appendix A - Measure and Integral

lim infi→∞

Ai =∞⋃

k=1

∞⋂i=k

Ai ∈ A; (A.3)

limi→∞

Ai ∈ A. (A.4)

Theorem A.2 The intersection of any collection of σ-algebras is a σ-algebra.Furthermore, for any nonempty class C, there is a unique minimal σ-algebracontaining C.

Theorem A.3 (Monotone Class Theorem) Assume that A0 is an algebraover Ω, and C is a monotone class of subsets of Ω (if Ai ∈ C and Ai ↑ Aor Ai ↓ A, then A ∈ C). If A0 ⊂ C and σ(A0) is the smallest σ-algebracontaining A0, then σ(A0) ⊂ C.

Definition A.2 Let Ω be a nonempty set, and A a σ-algebra over Ω. Then(Ω,A) is called a measurable space, and any elements in A are called mea-surable sets.

Theorem A.4 The smallest σ-algebra B containing all open intervals iscalled the Borel algebra of <, any elements in B are called a Borel set.

Borel algebra B is a special σ-algebra over <. It follows from Theorem A.2that Borel algebra is unique. It has been proved that open set, closed set,the set of rational numbers, the set of irrational numbers, and countable setof real numbers are all Borel sets.

Example A.3: We divide the interval [0, 1] into three equal open intervalsfrom which we choose the middle one, i.e., (1/3, 2/3). Then we divide eachof the remaining two intervals into three equal open intervals, and choosethe middle one in each case, i.e., (1/9, 2/9) and (7/9, 8/9). We perform thisprocess and obtain Dij for j = 1, 2, · · · , 2i−1 and i = 1, 2, · · · Note that Dijis a sequence of mutually disjoint open intervals. Without loss of generality,suppose Di1 < Di2 < · · · < Di,2i−1 for i = 1, 2, · · · Define the set

D =∞⋃

i=1

2i−1⋃j=1

Dij . (A.5)

Then C = [0, 1] \ D is called the Cantor set. In other words, x ∈ C if andonly if x can be expressed in ternary form using only digits 0 and 2, i.e.,

x =∞∑

i=1

ai

3i(A.6)

where ai = 0 or 2 for i = 1, 2, · · · The Cantor set is closed, uncountable, anda Borel set.

Let Ω1,Ω2, · · · ,Ωn be any sets (not necessarily subsets of the same space).The product Ω = Ω1 ×Ω2 × · · · ×Ωn is the set of all ordered n-tuples of theform (x1, x2, · · · , xn), where xi ∈ Ωi for i = 1, 2, · · · , n.

Section 1.2 - Classical Measures 231

Definition A.3 Let Ai be σ-algebras over Ωi, i = 1, 2, · · · , n, respectively.Write Ω = Ω1 × Ω2 × · · · × Ωn. A measurable rectangle in Ω is a set A =A1 × A2 × · · · × An, where Ai ∈ Ai for i = 1, 2, · · · , n. The smallest σ-algebra containing all measurable rectangles of Ω is called the product σ-algebra, denoted by A = A1 ×A2 × · · · ×An.

Note that the product σ-algebra A is the smallest σ-algebra containingmeasurable rectangles, rather than the product of A1,A2, · · · ,An.

Product σ-algebra may be easily extended to countably infinite case bydefining a measurable rectangle as a set of the form

A = A1 ×A2 × · · ·

where Ai ∈ Ai for all i and Ai = Ωi for all but finitely many i. The smallestσ-algebra containing all measurable rectangles of Ω = Ω1×Ω2× · · · is calledthe product σ-algebra, denoted by

A = A1 ×A2 × · · ·

A.2 Classical Measures

This appendix introduces the concepts of classical measure, measure space,Lebesgue measure, and product measure.

Definition A.4 Let Ω be a nonempty set, and A a σ-algebra over Ω. Aclassical measure π is a set function on A satisfyingAxiom 1. (Nonnegativity) πA ≥ 0 for any A ∈ A;Axiom 2. (Countable Additivity) For every countable sequence of mutuallydisjoint measurable sets Ai, we have

π

∞⋃i=1

Ai

=

∞∑i=1

πAi. (A.7)

Example A.4: Length, area and volume are instances of measure concept.

Definition A.5 Let Ω be a nonempty set, A a σ-algebra over Ω, and π ameasure on A. Then the triplet (Ω,A, π) is called a measure space.

It has been proved that there is a unique measure π on the Borel algebraof < such that π(a, b] = b− a for any interval (a, b] of <.

Definition A.6 The measure π on the Borel algebra of < satisfying

π(a, b] = b− a, ∀(a, b]

is called the Lebesgue measure.


Theorem A.5 (Product Measure Theorem) Let (Ωi,Ai, πi), i = 1, 2, · · · , nbe measure spaces such that πiΩi <∞ for i = 1, 2, · · · , n. Write

Ω = Ω1 × Ω2 × · · · × Ωn, A = A1 ×A2 × · · · ×An.

Then there is a unique measure π on A such that

πA1 ×A2 × · · · ×An = π1A1 × π2A2 × · · · × πnAn (A.8)

for every measurable rectangle A1 × A2 × · · · × An. The measure π is calledthe product of π1, π2, · · · , πn, denoted by π = π1 × π2 × · · · × πn. The triplet(Ω,A, π) is called the product measure space.

Theorem A.6 (Infinite Product Measure Theorem) Assume that (Ωi,Ai, πi)are measure spaces such that πiΩi = 1 for i = 1, 2, · · · Write

Ω = Ω1 × Ω2 × · · · A = A1 ×A2 × · · ·

Then there is a unique measure π on A such that

πA1×· · ·×An×Ωn+1×Ωn+2×· · · = π1A1×π2A2×· · ·×πnAn (A.9)

for any measurable rectangle A1 × · · · × An × Ωn+1 × Ωn+2 × · · · and alln = 1, 2, · · · The measure π is called the infinite product, denoted by π =π1×π2×· · · The triplet (Ω,A, π) is called the infinite product measure space.

A.3 Measurable Functions

This section introduces the concepts of measurable function, simple function,step function, absolutely continuous function, singular function, and Cantorfunction.

Definition A.7 A function f from (Ω,A) to the set of real numbers is saidto be measurable if

f−1(B) = ω ∈ Ω|f(ω) ∈ B ∈ A (A.10)

for any Borel set B of real numbers. If Ω is a Borel set, then A is alwaysassumed to be the Borel algebra over Ω.

Theorem A.7 The function f is measurable from (Ω,A) to < if and only iff−1(I) ∈ A for any open interval I of <.

Definition A.8 A function f : < → < is said to be continuous if for anygiven x ∈ < and ε > 0, there exists a δ > 0 such that |f(y) − f(x)| < εwhenever |y − x| < δ.

Section 1.3 - Measurable Functions 233

Example A.5: Any continuous function f is measurable, because f−1(I) isan open set (not necessarily interval) of < for any open interval I ∈ <.

Example A.6: A monotone function f from < to < is measurable becausef−1(I) is an interval for any interval I.

Example A.7: A function is said to be simple if it takes a finite set ofvalues. A function is said to be step if it takes a countably infinite set ofvalues. Generally speaking, a step (or simple) function is not necessarilymeasurable except that it can be written as f(x) = ai if x ∈ Ai, where Ai

are measurable sets, i = 1, 2, · · · , respectively.

Example A.8: Let f be a measurable function from (Ω,A) to <. Then itspositive part and negative part

f+(ω) =

f(ω), if f(ω) ≥ 0

0, otherwise,f−(ω) =

−f(ω), if f(ω) ≤ 0

0, otherwise

are measurable functions, becauseω∣∣ f+(ω) > t

=ω∣∣ f(ω) > t

∪ω∣∣ f(ω) ≤ 0 if t < 0

,

ω∣∣ f−(ω) > t

=ω∣∣ f(ω) < −t

∪ω∣∣ f(ω) ≥ 0 if t < 0

.

Example A.9: Let f1 and f2 be measurable functions from (Ω,A) to <.Then f1 ∨ f2 and f1 ∧ f2 are measurable functions, because

ω∣∣ f1(ω) ∨ f2(ω) > t

=ω∣∣ f1(ω) > t

∪ω∣∣ f2(ω) > t

,

ω∣∣ f1(ω) ∧ f2(ω) > t

=ω∣∣ f1(ω) > t

∩ω∣∣ f2(ω) > t

.

Theorem A.8 Let fi be a sequence of measurable functions from (Ω,A)to <. Then the following functions are measurable:

sup1≤i<∞

fi(ω); inf1≤i<∞

fi(ω); lim supi→∞

fi(ω); lim infi→∞

fi(ω). (A.11)

Especially, if limi→∞ fi(ω) exists, then it is also a measurable function.

Theorem A.9 Let f be a nonnegative measurable function from (Ω,A) to<. Then there exists an increasing sequence hi of nonnegative simple mea-surable functions such that

limi→∞

hi(ω) = f(ω), ∀ω ∈ Ω. (A.12)

Furthermore, the functions hi may be defined as follows,

hi(ω) =

k − 1

2i, if

k − 12i

≤ f(ω) <k

2i, k = 1, 2, · · · , i2i

i, if i ≤ f(ω)(A.13)

for i = 1, 2, · · ·


Definition A.9 A function f : < → < is said to be Lipschitz continuous ifthere is a positive number K such that

|f(y)− f(x)| ≤ K|y − x|, ∀x, y ∈ <. (A.14)

Definition A.10 A function f : < → < is said to be absolutely continuousif for any given ε > 0, there exists a small number δ > 0 such that

m∑i=1

|f(yi)− f(xi)| < ε (A.15)

for every finite disjoint class (xi, yi), i = 1, 2, · · · ,m of bounded open in-tervals for which

m∑i=1

|yi − xi| < δ. (A.16)

Definition A.11 A continuous and increasing function f : < → < is said tobe singular if f is not a constant and its derivative f ′ = 0 almost everywhere.

Example A.10: Let C be the Cantor set. We define a function g on C asfollows,

g

( ∞∑i=1

ai

3i

)=

∞∑i=1

ai

2i+1(A.17)

where ai = 0 or 2 for i = 1, 2, · · · Then g(x) is an increasing function andg(C) = [0, 1]. The Cantor function is defined on [0, 1] as follows,

f(x) = supg(y)

∣∣ y ∈ C, y ≤ x. (A.18)

It is clear that the Cantor function f(x) is increasing such that

f(0) = 0, f(1) = 1, f(x) = g(x), ∀x ∈ C.

Moreover, f(x) is a continuous function and f ′(x) = 0 almost everywhere.Thus the Cantor function f is a singular function.

A.4 Lebesgue Integral

This section introduces Lebesgue integral, Lebesgue-Stieltjes integral, mono-tone convergence theorem, Fatou’s lemma, Lebesgue dominated convergencetheorem, and Fubini theorem.

Definition A.12 Let h(x) be a nonnegative simple measurable function de-fined by

h(x) =

c1, if x ∈ A1

c2, if x ∈ A2

· · ·cm, if x ∈ Am

Section 1.4 - Lebesgue Integral 235

where A1, A2, · · · , Am are Borel sets. Then the Lebesgue integral of h on aBorel set A is ∫

A

h(x)dx =m∑

i=1

ciπA ∩Ai. (A.19)

Definition A.13 Let f(x) be a nonnegative measurable function on the Borelset A, and hi(x) a sequence of nonnegative simple measurable functionssuch that hi(x) ↑ f(x) as i→∞. Then the Lebesgue integral of f on A is∫

A

f(x)dx = limi→∞

∫A

hi(x)dx. (A.20)

Definition A.14 Let f(x) be a measurable function on the Borel set A, anddefine

f+(x) =

f(x), if f(x) > 0

0, otherwise,f−(x) =

−f(x), if f(x) < 0

0, otherwise.

Then the Lebesgue integral of f on A is∫A

f(x)dx =∫

A

f+(x)dx−∫

A

f−(x)dx (A.21)

provided that at least one of∫

Af+(x)dx and

∫Af−(x)dx is finite.

Definition A.15 Let f(x) be a measurable function on the Borel set A. Ifboth of

∫Af+(x)dx and

∫Af−(x)dx are finite, then the function f is said to

be integrable on A.

Theorem A.10 (Monotone Convergence Theorem) Let fi be an increas-ing sequence of measurable functions on A. If there is an integrable functiong such that fi(x) ≥ g(x) for all i, then we have∫

A

limi→∞

fi(x)dx = limi→∞

∫A

fi(x)dx. (A.22)

Example A.11: The condition fi ≥ g cannot be removed in the monotoneconvergence theorem. For example, let fi(x) = 0 if x ≤ i and −1 otherwise.Then fi(x) ↑ 0 everywhere on A. However,∫

<lim

i→∞fi(x)dx = 0 6= −∞ = lim

i→∞

∫<fi(x)dx.

Theorem A.11 (Fatou’s Lemma) Assume that fi is a sequence of mea-surable functions on A. (a) If there is an integrable function g such thatfi ≥ g for all i, then∫

A

lim infi→∞

fi(x)dx ≤ lim infi→∞

∫A

fi(x)dx. (A.23)


(b) If there is an integrable function g such that fi ≤ g for all i, then∫A

lim supi→∞

fi(x)dx ≥ lim supi→∞

∫A

fi(x)dx. (A.24)

Theorem A.12 (Lebesgue Dominated Convergence Theorem) Let fi be asequence of measurable functions on A whose limitation limi→∞ fi(x) existsa.s. If there is an integrable function g such that |fi(x)| ≤ g(x) for any i,then we have ∫

A

limi→∞

fi(x)dx = limi→∞

∫A

fi(x)dx. (A.25)

Example A.12: The condition |fi| ≤ g in the Lebesgue dominated conver-gence theorem cannot be removed. Let A = (0, 1), fi(x) = i if x ∈ (0, 1/i)and 0 otherwise. Then fi(x) → 0 everywhere on A. However,∫

A

limi→∞

fi(x)dx = 0 6= 1 = limi→∞

∫A

fi(x)dx.

Theorem A.13 (Fubini Theorem) Let f(x, y) be an integrable function on<2. Then we have(a) f(x, y) is an integrable function of x for almost all y;(b) f(x, y) is an integrable function of y for almost all x;

(c)∫<2f(x, y)dxdy =

∫<

(∫<f(x, y)dy

)dx =

∫<

(∫<f(x, y)dx

)dy.

Theorem A.14 Let Φ be an increasing and right-continuous function on <.Then there exists a unique measure π on the Borel algebra of < such that

π(a, b] = Φ(b)− Φ(a) (A.26)

for all a and b with a < b. Such a measure is called the Lebesgue-Stieltjesmeasure corresponding to Φ.

Definition A.16 Let Φ(x) be an increasing and right-continuous functionon <, and let h(x) be a nonnegative simple measurable function, i.e.,

h(x) =

c1, if x ∈ A1

c2, if x ∈ A2

...cm, if x ∈ Am.

Then the Lebesgue-Stieltjes integral of h on the Borel set A is∫A

h(x)dΦ(x) =m∑

i=1

ciπA ∩Ai (A.27)

where π is the Lebesgue-Stieltjes measure corresponding to Φ.

Section 1.4 - Lebesgue Integral 237

Definition A.17 Let f(x) be a nonnegative measurable function on the Borelset A, and let hi(x) be a sequence of nonnegative simple measurable func-tions such that hi(x) ↑ f(x) as i → ∞. Then the Lebesgue-Stieltjes integralof f on A is ∫

A

f(x)dΦ(x) = limi→∞

∫A

hi(x)dΦ(x). (A.28)

Definition A.18 Let f(x) be a measurable function on the Borel set A, anddefine

f+(x) =

f(x), if f(x) > 0

0, otherwise,f−(x) =

−f(x), if f(x) < 0

0, otherwise.

Then the Lebesgue-Stieltjes integral of f on A is∫A

f(x)dΦ(x) =∫

A

f+(x)dΦ(x)−∫

A

f−(x)dΦ(x) (A.29)

provided that at least one of∫

Af+(x)dΦ(x) and

∫Af−(x)dΦ(x) is finite.

Appendix B

Euler-Lagrange Equation

Let φ(x) be an unknown function. We consider the functional of φ as follows,

L(φ) =∫ b

a

F (x, φ(x), φ′(x))dx (B.1)

where F is a known function with continuous first and second partial deriva-tives. If L has an extremum (maximum or minimum) at φ(x), then

∂F

∂φ(x)− d

dx

(∂F

∂φ′(x)

)= 0 (B.2)

which is called the Euler-Lagrange equation. If φ′(x) is not involved, thenthe Euler-Lagrange equation reduces to

∂F

∂φ(x)= 0. (B.3)

Note that the Euler-Lagrange equation is only a necessary condition for theexistence of an extremum. However, if the existence of an extremum is clearand there exists only one solution to the Euler-Lagrange equation, then thissolution must be the curve for which the extremum is achieved.

Appendix C

Logic

Propositions, connective symbols, formulas, truth functions, truth value arebasic concepts in logic. This appendix also introduces the probabilistic logic,credibilistic logic and hybrid logic.

C.1 Traditional Logic

Propositions

A proposition is a statement with truth value belonging to 0, 1.

Connective Symbols

In addition to the proposition symbols ξ and η, we also need the negationsymbol ¬, conjunction symbol ∧, disjunction symbol ∨, and implication sym-bol →. Note that ¬,∨ is complete, and

ξ ∧ η = ¬(¬ξ ∨ ¬η), ξ → η = (¬ξ) ∨ η. (C.1)

Formulas

A formula is a finite sequence of proposition and connective symbols thatmust make sense. In other words, the nonsensical sequence must be excluded.It is clear that any propositions ξ, η and τ are formulas, and so are

¬ξ, ξ ∨ η, ξ ∧ η, ξ → η, (ξ ∨ η) → τ.

However, ¬ ∨ ξ, ξ → ∨ and ξη → τ are not formulas. For any formula X, ifX = 1, then X is true; if X = 0, then X is false.

240 Appendix C - Logic

Truth Functions

AssumeX is a formula containing propositions ξ1, ξ2, · · · , ξn. It is well-knownthat there is a function f : 0, 1n → 0, 1 such that X = 1 if and only iff(ξ1, ξ2, · · · , ξn) = 1. Such a Boolean function f is called the truth functionof X. For example, the truth function of formula ξ1 ∨ ξ2 → ξ3 is

f(1, 1, 1) = 1, f(1, 0, 1) = 1, f(0, 1, 1) = 1, f(0, 0, 1) = 1,f(1, 1, 0) = 0, f(1, 0, 0) = 0, f(0, 1, 0) = 0, f(0, 0, 0) = 1.

C.2 Probabilistic Logic

Probabilistic logic was proposed by Nilsson [186] as a generalization of logicfor dealing with random knowledge. A random proposition is a statement withprobability value belonging to [0, 1]. A random formula is a finite sequence ofrandom propositions and connective symbols that must make sense. Let Xbe a random formula. Then the truth value of X is defined as the probabilitythat the random formula X is true, i.e.,

T (X) = PrX = 1. (C.2)

If X is a random formula containing random propositions ξ1, ξ2, · · · , ξn whosetruth function is f(x1, x2, · · · , xn), then the truth value of X is

T (X) =∑

f(x1,x2,··· ,xn)=1

p(x1, x2, · · · , xn) (C.3)

where p(x1, x2, · · · , xn) is the joint probability mass function of (ξ1, ξ2, · · · , ξn).

Example C.1: Let ξ and η be independent random propositions with prob-ability values a and b, respectively. Then we have(a) T (ξ) = Prξ = 1 = a;(b) T (¬ξ) = Prξ = 0 = 1− a;(c) T (ξ ∧ η) = Prξ ∧ η = 1 = Pr(ξ = 1) ∩ (η = 1) = a · b;(d) T (ξ ∨ η) = Prξ ∨ η = 1 = Pr(ξ = 1) ∪ (η = 1) = a+ b− a · b;(e) T (ξ → η) = Pr(ξ → η) = 1 = Pr(ξ = 0) ∪ (η = 1) = 1− a+ a · b.

Remark C.1: Probabilistic logic is consistent with the law of contradictionand law of excluded middle, i.e.,

T (X ∧ ¬X) = 0, T (X ∨ ¬X) = 1, T (X) + T (¬X) = 1. (C.4)

C.3 Credibilistic Logic

Credibilistic logic was designed by Li and Liu [109] as a generalization oflogic for dealing with fuzzy knowledge. A fuzzy proposition is a statementwith credibility value belonging to [0, 1]. A fuzzy formula is a finite sequence

Section 3.4 - Hybrid Logic 241

of fuzzy propositions and connective symbols that must make sense. Let Xbe a fuzzy formula. Then the truth value of X is defined as the credibilitythat the fuzzy formula X is true, i.e.,

T (X) = CrX = 1. (C.5)

If X is a fuzzy formula containing fuzzy propositions ξ1, ξ2, · · · , ξn whosetruth function is f(x1, x2, · · · , xn), then the truth value of X is

T (X) =12

(max

f(x)=1µ(x) + 1− max

f(x)=0µ(x)

)(C.6)

where x = (x1, x2, · · · , xn), and µ(x) is the joint membership function of(ξ1, ξ2, · · · , ξn).

Example C.2: Let ξ and η be (not necessarily independent) fuzzy proposi-tions with credibility values a and b, respectively. Then we have(a) T (ξ) = Crξ = 1 = a;(b) T (¬ξ) = Crξ = 0 = 1− a;(c) T (ξ ∧ η) = Cr(ξ = 1) ∩ (η = 1) = a ∧ b provided that a+ b > 1;(d) T (ξ ∨ η) = Cr(ξ = 1) ∪ (η = 1) = a ∨ b provided that a+ b < 1;(e) T (ξ → η) = Cr(ξ = 0) ∪ (η = 1) = (1− a) ∨ b provided that a > b.

Example C.3: Let ξ and η be independent fuzzy propositions with credi-bility values a and b, respectively. Then we have(a) T (ξ ∧ η) = Cr(ξ = 1) ∩ (η = 1) = a ∧ b;(b) T (ξ ∨ η) = Cr(ξ = 1) ∪ (η = 1) = a ∨ b;(c) T (ξ → η) = Cr(ξ = 0) ∪ (η = 1) = (1− a) ∨ b.

Remark C.2: Credibilistic logic is consistent with the law of contradictionand law of excluded middle, i.e.,

T (X ∧ ¬X) = 0, T (X ∨ ¬X) = 1, T (X) + T (¬X) = 1. (C.7)

This is the key point that credibilistic logic is different from possibilistic logic.

C.4 Hybrid Logic

Hybrid logic was designed by Li and Liu [108] in 2009 as a generalizationof logic for dealing with fuzzy knowledge and random knowledge simultane-ously. A hybrid formula is a finite sequence of fuzzy propositions, randompropositions and connective symbols that must make sense. Let X be a hy-brid formula. Then the truth value of X is defined as the chance that thehybrid formula X is true, i.e.,

T (X) = ChX = 1. (C.8)

242 Appendix C - Logic

If X is a hybrid formula containing fuzzy propositions ξ1, ξ2, · · · , ξn and ran-dom propositions η1, η2, · · · , ηm whose truth function is f , then the truthvalue of X is

T (X) =

supx

µ(x)2

∧∑

f(x,y)=1

φ(y)

,

if supx

µ(x)2

∧∑

f(x,y)=1

φ(y)

< 0.5

1− supx

µ(x)2

∧∑

f(x,y)=0

φ(y)

,

if supx

µ(x)2

∧∑

f(x,y)=1

φ(y)

≥ 0.5

(C.9)

where x = (x1, x2, · · · , xn), y = (y1, y2, · · · , ym), µ(x) is the joint mem-bership function of (ξ1, ξ2, · · · , ξn), and φ(y) is the joint probability massfunction of (η1, η2, · · · , ηm).

Example C.4: Let ξ, κ be independent fuzzy propositions with credibilityvalues a, c, and let η, τ be independent random propositions with probabilityvalues b, d, respectively. Then we have(a) T (ξ ∧ η) = Ch(ξ = 1) ∩ (η = 1) = a ∧ b;(b) T (ξ ∨ η) = Ch(ξ = 1) ∪ (η = 1) = a ∨ b;(c) T (ξ → τ) = Ch(ξ = 0) ∪ (τ = 1) = (1− a) ∨ d;(d) T (η → κ) = Ch(η = 0) ∪ (κ = 1) = (1− b) ∨ c.Furthermore, we have

T (ξ ∧ η → κ) = Ch(ξ ∧ η = 0) ∪ (κ = 1)= 1− Ch(ξ = 1) ∩ (η = 1) ∩ (κ = 0)= 1− Cr(ξ = 1) ∩ (κ = 0) ∧ Prη = 1= 1− a ∧ (1− c) ∧ b;

T (ξ ∧ η → τ) = Ch(ξ ∧ η = 0) ∪ (τ = 1)= 1− Ch(ξ = 1) ∩ (η = 1) ∩ (τ = 0)= 1− Crξ = 1 ∧ Pr(η = 1) ∩ (τ = 0)= 1− a ∧ (b− bd).

Remark C.3: Hybrid logic is consistent with the law of contradiction andlaw of excluded middle, i.e.,

T (X ∧ ¬X) = 0, T (X ∨ ¬X) = 1, T (X) + T (¬X) = 1. (C.10)

Section 3.5 - Uncertain Logic 243

C.5 Uncertain Logic

Uncertain logic was designed by Li and Liu [108] in 2009 as a generalizationof logic for dealing with uncertain knowledge. An uncertain proposition isa statement with truth value belonging to [0, 1]. An uncertain formula is afinite sequence of uncertain propositions and connective symbols that mustmake sense. Let X be an uncertain formula. Then the truth value of X isdefined as the uncertain measure that the uncertain formula X is true, i.e.,

T (X) = MX = 1. (C.11)

If X is an uncertain formula containing independent uncertain propositionsξ1, ξ2, · · · , ξn whose truth function is f , then the truth value of X is

T (X) =

supf(x1,x2,··· ,xn)=1

min1≤i≤n

νi(xi),


min1≤i≤n

νi(xi) < 0.5

1− supf(x1,x2,··· ,xn)=0

min1≤i≤n

νi(xi),


min1≤i≤n

νi(xi) ≥ 0.5

(C.12)

where νi(xi) are identification functions (no matter what types they are) ofξi, i = 1, 2, · · · , n, respectively.

Remark C.4: Uncertain logic is consistent with the law of contradictionand law of excluded middle, i.e.,

T (X ∧ ¬X) = 0, T (X ∨ ¬X) = 1, T (X) + T (¬X) = 1. (C.13)

Appendix D

Inference

This appendix will introduce traditional inference, fuzzy inference and un-certain inference.

D.1 Traditional Inference

As traditional rules of inference, the modus ponens, modus tollens and hypo-thetical syllogism play an important role in classical logic.

Modus Ponens: From formulas X and X → Y we may infer Y . That is, ifX and X → Y are true, then Y is true.

Modus Tollens: From formulas ¬Y and X → Y we may infer ¬X. Thatis, if Y is false and X → Y is true, then X is false.

Hypothetical Syllogism: From X → Y and Y → Z we infer X → Z.That is, if both X → Y and Y → Z are true, then X → Z is true.

D.2 Fuzzy Inference

Before introducing fuzzy inference, let us recall the concept of fuzzy set thatwas given by L.A. Zadeh in 1965 via membership function. For our purpose,we accept the following definition of fuzzy set.

Definition D.1 (Liu [138]) A fuzzy set is a function ξ from a credibilityspace (Θ,P,Cr) to a collection of sets of real numbers.

Let ξ and ξ∗ be two fuzzy sets. In order to define the degree that ξ∗

matches ξ, we first introduce the following symbols:

ξ∗ ⊂ ξ = θ ∈ Θ∣∣ ξ∗(θ) ⊂ ξ(θ), (D.1)

ξ∗ ⊂ ξc = θ ∈ Θ∣∣ ξ∗(θ) ⊂ ξ(θ)c, (D.2)

Section 4.3 - Hybrid Inference 245

ξ∗ 6⊂ ξc = θ ∈ Θ∣∣ ξ∗(θ) ∩ ξ(θ) 6= ∅. (D.3)

Definition D.2 Let ξ and ξ∗ be two fuzzy sets. Then the event ξ∗ B ξ (i.e.,ξ∗ matches ξ) is defined as

ξ∗ B ξ =

ξ∗ ⊂ ξ, if Crξ∗ ⊂ ξ > Crξ∗ ⊂ ξc

ξ∗ 6⊂ ξc, if Crξ∗ ⊂ ξ ≤ Crξ∗ ⊂ ξc.(D.4)

That is, we take the event of ξ∗ ⊂ ξ or ξ∗ 6⊂ ξc such that its credibilitymeasure is as closed to 0.5 as possible. This idea coincides with the maximumuncertainty principle.

Definition D.3 Let ξ and ξ∗ be two fuzzy sets. Then the matching degreeof ξ∗ B ξ is defined as

Crξ∗ B ξ =

Crξ∗ ⊂ ξ, if Crξ∗ ⊂ ξ > Crξ∗ ⊂ ξc

Crξ∗ 6⊂ ξc, if Crξ∗ ⊂ ξ ≤ Crξ∗ ⊂ ξc.(D.5)

Fuzzy inference has been studied by many scholars and a lot of inferencerules have been suggested. For example, Zadeh’s compositional rule of infer-ence, Lukasiewicz’s inference rule, and Mamdani’s inference rule are widelyused in fuzzy inference systems.

However, instead of using the above inference rules, we will introduce aninference rule proposed by Liu [141]. Let X and Y be two concepts. It isassumed that we have a rule “if X is a fuzzy set ξ then Y is a fuzzy set η”.From X is a fuzzy set ξ∗ we infer that Y is a fuzzy set

η∗ = η|ξ∗Bξ (D.6)

which is the conditional fuzzy set of η given ξ∗ B ξ.

D.3 Hybrid Inference

Definition D.4 (Liu [138]) A hybrid set is a function ξ from a chance spacespace (Θ,P,Cr) × (Ω,A,Pr) to a collection of sets of real numbers, i.e., forany Borel set B, the set

(θ, ω) ∈ Θ× Ω∣∣ ξ(θ, ω) ∩B 6= ∅ (D.7)

is an event.

Definition D.5 Let ξ and ξ∗ be two hybrid sets. Then the event ξ∗Bξ (i.e.,ξ∗ matches ξ) is defined as

ξ∗ B ξ =

ξ∗ ⊂ ξ, if Chξ∗ ⊂ ξ > Chξ∗ ⊂ ξc

ξ∗ 6⊂ ξc, if Chξ∗ ⊂ ξ ≤ Chξ∗ ⊂ ξc.(D.8)

246 Appendix D - Inference

Definition D.6 Let ξ and ξ∗ be two hybrid sets. Then the matching degreeof ξ∗ B ξ is defined as

Chξ∗ B ξ =

Crξ∗ ⊂ ξ, if Chξ∗ ⊂ ξ > Chξ∗ ⊂ ξc

Chξ∗ 6⊂ ξc, if Chξ∗ ⊂ ξ ≤ Chξ∗ ⊂ ξc.(D.9)

This section will introduce an inference rule proposed by Liu [141]. LetX and Y be two concepts. It is assumed that we have a rule “if X is a hybridset ξ then Y is a hybrid set η”. From X is a hybrid set ξ∗ we infer that Y isa hybrid set

η∗ = η|ξ∗Bξ (D.10)

which is the conditional hybrid set of η given ξ∗ B ξ.

D.4 Uncertain Inference

Uncertain inference was proposed by Liu [141] in 2009. Let X and Y be twoconcepts. It is assumed that we only have a rule “if X is an uncertain set ξthen Y is an uncertain set η”. From X is an uncertain set ξ∗, we infer thatY is an uncertain set

η∗ = η|ξ∗Bξ (D.11)

which is the conditional uncertain set of η given ξ∗ B ξ. Thus we have thefollowing inference rule:

Rule: If X is ξ then Y is ηFrom: X is ξ∗

Infer: Y is η∗ = η|ξ∗Bξ

(D.12)

Appendix E

Maximum UncertaintyPrinciple

An event has no uncertainty if its measure is 1 (or 0) because we may believethat the event occurs (or not). An event is the most uncertain if its measureis 0.5 because the event and its complement may be regarded as “equallylikely”.

In practice, if there is no information about the measure of an event,we should assign 0.5 to it. Sometimes, only partial information is available.For this case, the value of measure may be specified in some range. Whatvalue does the measure take? For the safety purpose, we should assign it thevalue as close to 0.5 as possible. This is the maximum uncertainty principleproposed by Liu [138].

Maximum Uncertainty Principle: For any event, if there are multiplereasonable values that a measure may take, then the value as close to 0.5 aspossible is assigned to the event.

Perhaps the reader would like to ask what values are reasonable. Theanswer is problem-dependent. At least, the values should ensure that allaxioms about the measure are satisfied, and should be consistent with thegiven information.

Example E.1: Let Λ be an event. Based on some given information, themeasure value MΛ is on the interval [a, b]. By using the maximum uncer-tainty principle, we should assign

MΛ =

a, if 0.5 < a ≤ b

0.5, if a ≤ 0.5 ≤ b

b, if a ≤ b < 0.5.

Appendix F

Why Uncertain Measure?

Classical measure and probability measure belong to the class of “completelyadditive measure”.

Capacity, belief measure, plausibility measure, fuzzy measure, possibilitymeasure and necessity measure belong to the class of “completely nonaddi-tive measure”. Since this class of measures does not assume the self-dualityproperty, all of those measures are inconsistent with the law of contradictionand law of excluded middle.

Uncertain measure is neither a completely additive measure nor a com-pletely nonadditive measure. In fact, uncertain measure is a “partially ad-ditive measure” because of its self-duality. Credibility measure and chancemeasure are special types of uncertain measure. One advantage of this classof measures is the well consistency with the law of contradiction and law ofexcluded middle.

Completely Classical Measure (Borel and Lebesgue, 1900)Additive Measures Probability Measure (Kolmogoroff, 1933)

Capacity (Choquet, 1954)Belief Measure (Dempster, 1967; Shafer, 1976)

Completely Plausibility Measure (ibid)Nonadditive Measures Fuzzy Measure (Sugeno, 1974)

Possibility Measure (Zadeh, 1978)Necessity Measure (Zadeh, 1979)

Uncertain Measure Credibility Measure (Liu and Liu, 2002)(Liu, 2007) Chance Measure (Li and Liu, 2009)

The essential differentia between fuzziness and uncertainty is that theformer assumes

MA ∪B = MA ∨MBfor any events A and B no matter if they are independent or not, and thelatter assumes the relation only for independent events. However, a lot of

Appendix F - Why Uncertain Measure? 249

surveys showed that

MA ∪B 6= MA ∨MB

when A and B are not independent events. Fuzziness is an extremely specialbut non-practical case of uncertainty. This is the main reason why we needthe uncertainty theory.

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List of Frequently Used Symbols

ξ, η, τ random, fuzzy, hybrid, or uncertain variablesξ, η, τ random, fuzzy, hybrid, or uncertain vectorsµ, ν membership functionsφ, ψ probability, or credibility density functionsΦ, Ψ probability, or credibility distributions

Pr probability measureCr credibility measureCh chance measureM uncertain measureE expected valueV varianceH entropy

(Γ,L,M) uncertainty space(Ω,A,Pr) probability space(Θ,P,Cr) credibility space

(Θ,P,Cr)× (Ω,A,Pr) chance space∅ empty set< set of real numbers<n set of n-dimensional real vectors∨ maximum operator∧ minimum operator

ai ↑ a a1 ≤ a2 ≤ · · · and ai → a

ai ↓ a a1 ≥ a2 ≥ · · · and ai → a

Ai ↑ A A1 ⊂ A2 ⊂ · · · and A = A1 ∪A2 ∪ · · ·Ai ↓ A A1 ⊃ A2 ⊃ · · · and A = A1 ∩A2 ∩ · · ·

Index

algebra, 229arithmetic canonical process, 206bell-like uncertain variable, 167Borel algebra, 230Borel set, 230Brownian motion, 205canonical process, 205Cantor function, 234Cantor set, 230chance asymptotic theorem, 124chance density function, 131chance distribution, 130chance measure, 117chance semicontinuity law, 123chance space, 115chance subadditivity theorem, 121Chebyshev inequality, 33, 102, 141conditional chance, 148conditional credibility, 108conditional fuzzy variable, 111conditional random variable, 151conditional membership function, 110conditional probability, 40conditional random variable, 41conditional uncertain measure, 196conditional uncertain variable, 198connective symbols, 239control system, 216convergence almost surely, 35, 104convergence in chance, 143convergence in credibility, 104convergence in distribution, 36, 105convergence in mean, 36, 104convergence in probability, 35countable additivity axiom, 1countable subadditivity axiom, 153credibilistic logic, 240credibility asymptotic theorem, 50credibility density function, 66

credibility distribution, 62credibility extension theorem, 51credibility inversion theorem, 59credibility measure, 45credibility semicontinuity law, 49credibility space, 52credibility subadditivity theorem, 47critical value, 26, 90, 137, 187distance, 32, 100, 140, 190entropy, 28, 94, 188equipossible fuzzy variable, 61Euler-Lagrange equation, 238event, 1, 45, 116, 153expected value, 14, 73, 132, 176exponential distribution, 10exponential membership function, 89exponential uncertain variable, 167extension principle of Zadeh, 72first identification function, 164Fubini theorem, 236fuzzy inference, 244fuzzy random variable, 115fuzzy sequence, 104fuzzy set, 244fuzzy variable, 57fuzzy vector, 57geometric canonical process, 207hazard rate, 43, 112Holder’s inequality, 34, 102, 141, 192hybrid inference, 245hybrid logic, 241hybrid sequence, 143hybrid set, 245hybrid variable, 124hybrid vector, 128hypothetical syllogism, 244identification function, 164independence, 11, 68, 173inference rule, 226

Index 269

Jensen’s inequality, 35, 103, 142, 193law of contradiction, 154, 221law of excluded middle, 154, 221law of truth conservation, 222Lebesgue integral, 234Lebesgue measure, 231Lebesgue-Stieltjes integral, 236Lebesgue-Stieltjes measure, 236Lebesgue unit interval, 4Markov inequality, 33, 102, 141, 192matching degree, 226, 245maximality axiom, 46maximum entropy principle, 30, 97maximum uncertainty principle, 247measurable function, 232measure inversion theorem, 169membership function, 58Minkowski inequality, 34, 103, 142modus ponens, 244modus tollens, 244moment, 25, 89, 137, 185monotone convergence theorem, 235monotone class theorem, 230monotonicity axiom, 45, 153nonnegativity axiom, 1normal membership function, 87normal probability distribution, 10normal uncertain variable, 183normality axiom, 1, 45, 153operational law, 175optimistic value, see critical valueoption pricing, 213pessimistic value, see critical valueportfolio selection, 215power set, 229probabilistic logic, 240probability continuity theorem, 2probability density function, 9probability distribution, 7probability inversion theorem, 9probability measure, 3probability space, 3product measure axiom, 159product credibility axiom, 54product credibility space, 55

product credibility theorem, 54product probability space, 4product probability theorem, 4product uncertainty space, 161random fuzzy variable, 115random sequence, 35random variable, 5random vector, 5rectangular uncertain variable, 165renewal process, 202second identification function, 166self-duality axiom, 45, 153set function, 1σ-algebra, 229simple function, 233singular function, 234step function, 233stock model, 213trapezoidal fuzzy variable, 61trapezoidal uncertain variable, 165triangular fuzzy variable, 61triangular uncertain variable, 165third identification function, 169truth function, 240truth value, 220uncertain calculus, 205uncertain control, 216uncertain differential equation, 211uncertain dynamical system, 217uncertain filtering, 216uncertain finance, 213uncertain inference, 225uncertain logic, 219uncertain measure, 154uncertain process, 201uncertain sequence, 194uncertain set, 225uncertain variable, 162uncertain vector, 162uncertainty asymptotic theorem, 158uncertainty density function, 172uncertainty distribution, 170uncertainty space, 158uniform distribution, 10

variance, 23, 86, 135, 183

Axiom 1. (Normality) MΓ = 1 for the universal set Γ.

Axiom 2. (Monotonicity) MΛ1 ≤MΛ2 whenever Λ1 ⊂ Λ2.

Axiom 3. (Self-Duality) MΛ+MΛc = 1 for any event Λ.

Axiom 4. (Countable Subadditivity) For every countable sequence of eventsΛ1,Λ2, · · · , we have

M

∞⋃i=1

Λi

≤

∞∑i=1

MΛi.

Axiom 5. (Product Measure) Let (Γk,Lk,Mk) be uncertainty spaces fork = 1, 2, · · · , n. The product uncertain measure is M = M1 ∧M2 ∧ · · · ∧Mn,i.e.,

MΛ =


min1≤k≤n

MkΛk,

if supΛ1×Λ2×···×Λn⊂Λ

min1≤k≤n

MkΛk > 0.5

1− supΛ1×Λ2×···×Λn⊂Λc

min1≤k≤n

MkΛk,

if supΛ1×Λ2×···×Λn⊂Λc

min1≤k≤n

MkΛk > 0.5

0.5, otherwise.

Baoding LiuUncertainty Theory

Uncertainty theory is a branch of mathematics based on normality, mono-tonicity, self-duality, countable subadditivity, and product measure axioms.The goal of uncertainty theory is to study the behavior of subjective un-certainty. For this new edition the entire text has been totally rewritten.More importantly, uncertain process, uncertain calculus, uncertain differen-tial equation, uncertain logic and uncertain inference are completely new.This book provides a self-contained, comprehensive and up-to-date presen-tation of uncertainty theory. The purpose is to equip the readers with anaxiomatic approach to deal with subjective uncertainty. Mathematicians,researchers, engineers, designers, and students in the field of mathematics,information science, operations research, industrial engineering, computerscience, artificial intelligence, and management science will find this work astimulating and useful reference.

Documents

Uncertainty Theory, 3rd ed., 2009; Liu B