Probability in Petroleum and Environmental Engineering

Probability in Petroleum and Environmental Engineering/11304_fm.pdfProbability inPetroleum andEnvironmentalEngineering

Leonid R KhilyukConsultant in Mathematical Modeling,Los Angeles, California, USA

George V. ChilingarSchool of Engineering,University of Southern California,Los Angeles, California, USA

Herman H, RiekePetroleum Engineering Department,University of Louisiana at Lafayette,Lafayette, Louisiana, USA

Houston, Texas

Copyright 2005 by Gulf Publishing Company Houston, Texas.AU rights reserved. No part of this publication may be reproducedor transmitted in any form without the prior written permissionof the publisher.

Gulf Publishing Company2 Greenway Plaza, Suite 1020Houston, TX 77046

10 9 8 7 6 5 4 3 2 1

Printed in the United States of America.

Printed on acid-free paper.

Text design and composition by Ruth Maassen.

ISBN 0-9765113-0-4

This book is dedicated toHis Majesty King Bhumibol Adulyadej

and Her Majesty Queen Sirikit of Thailandfor their relentless efforts

to raise the standard of livingof their wonderful people.

PREFACE

From thermodynamics to quantum mechanics to modern commu-nication technology, important concepts from probability and sta-tistics have become increasingly dominant in many areas of scienceand engineering over the past 150 years. This book plays a vital roleof introducing the fundamental ideas of probability and statistics ina way that is directly applicable to practical situations in environ-mental and petroleum engineering, and also is sound mathemati-cally. For example, estimation of pollution levels in air, water, andsoil, as well as the characterization of petroleum reservoirs, areappropriately illustrated in a probabilistic context. Statistical testingand measures of significance are explained clearly, and each chapterconcludes with an excellent set of questions and exercises.

Solomon W. Golomb, Academician,University Professor of Engineering and Mathematics,

University of Southern California, Los Angeles

LIST OF NOTATIONS

Symbol Meaning

A=> B Statement B follows from statement A;A implies B.

A Y fis a function with domain X and range Y.

Customary notation is y = fix), xeX,ye Y.xeA x is an element of set A.x & A x is not an element of set A.A Kj B Denotes union of two sets A and B.AnB Denotes intersection of two sets A and B.A czB A is a subset of B,A=B A and B are equal if A aBmdBcA.0 Denotes empty set.

Front MatterPrefaceList of Notations

Table of ContentsIndex

Probability in Petroleum and Environmental Engineering/Thumbs.dbProbability in Petroleum and Environmental Engineering/11304_toc.pdfvii This page has been reformatted by Knovel to provide easier navigation.

Contents

Preface ............................................................................ xiii

List of Notations ............................................................... xv

1. Introduction ............................................................. 1 1.1 The Approach ........................................................... 1 1.2 Overview ................................................................... 4 1.3 Instructions ................................................................ 7

2. Experiments and Events ........................................ 9 2.1 Primary Notions ........................................................ 9 2.2 Algebra of Events ..................................................... 12 2.3 Relation of Implication .............................................. 13 2.4 Main Operations with Events .................................... 15 2.5 Main Properties of the Operations with

Events ....................................................................... 18 2.6 Theorem on the Decomposition of an Event

into a Complete Set of Events .................................. 19 2.7 Interpretation of Environmental Phenomena as

Events of Experiments .............................................. 20 2.8 Questions and Exercises .......................................... 23

3. Space of Elementary Events .................................. 25 3.1 Preliminary Remarks ................................................ 25 3.2 Composition of the Space of Elementary

Events ....................................................................... 31

viii Contents

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3.3 Composition of the Space of Elementary Events for Air-quality Monitoring and Forecasting ............... 34

3.4 Characterization of the Eutrophication of a Bay Water ........................................................................ 37

3.5 Questions and Exercises .......................................... 38

4. Probability of Random Events ............................... 41 4.1 Random Events and Random Experiments ............. 41 4.2 The Concept of Probability of a Random

Event ......................................................................... 42 4.3 Adequacy of Chosen Probabilistic Space to the

Given Stochastic Experiment ................................... 43 4.4 Corollaries of Probability Axioms .............................. 45 4.5 Classic Definition of Probability ................................ 47 4.6 Geometric Definition of Probability ........................... 51 4.7 Statistical Definition of Probability ............................ 56 4.8 Questions and Exercises .......................................... 58

5. Conditional Probability and Stochastic Independence: Multistage Probabilistic Evaluation and Forecasting ................................... 61 5.1 Conditional Probability .............................................. 61 5.2 Formula of Total Probability ..................................... 64 5.3 Bayes Formula ......................................................... 64 5.4 Examples of Application ........................................... 65 5.5 Independence of Events ........................................... 68 5.6 Multistage Probabilistic Assessment of

Failure ....................................................................... 70 5.7 Simplified Probabilistic Model for Air-quality

Forecasting ............................................................... 71 5.8 Probability of a Water-purification System

Being Functional ....................................................... 73 5.9 Questions and Exercises .......................................... 73

Contents ix


6. Bernoulli Distribution and Sequences of Independent Trials .................................................. 75 6.1 Bernoulli (Binomial) Distribution ............................... 75 6.2 Sequence of Independent Trials and Its

Mathematical Model ................................................. 76 6.3 Probabilistic Space for a Sequence of

Independent Experiments ........................................ 80 6.4 Bernoulli Scheme of Independent Trials .................. 80 6.5 Examples of Application ........................................... 82 6.6 Application of the Bernoulli Scheme for Air-quality

Assessment .............................................................. 85 6.7 Questions and Exercises .......................................... 86

7. Random Variables and Distribution Functions ................................................................ 89 7.1 Quantities Depending on Random Events ............... 89 7.2 Mathematical Definition of a Random

Variable ..................................................................... 90 7.3 Events Defined by Random Variables ..................... 91 7.4 Independent Random Variables ............................... 92 7.5 Distribution of a Random Variable: the

Distribution Function ................................................. 93 7.6 General Properties of Distribution Functions ........... 93 7.7 Discrete Random Variables ...................................... 95 7.8 Continuous Random Variables ................................ 98 7.9 General Properties of Distribution Density ............... 98 7.10 Distribution Function and Distribution Density

of Functions of Random Variables ........................... 102 7.11 Evaluating Probability of Soil and Groundwater

Contamination ........................................................... 105 7.12 Questions and Exercises .......................................... 108

x Contents


8. Numerical Characteristics of Random Variables: Mathematical Expectation, Variance, and Moments of Higher Order .............. 111 8.1 Introduction ............................................................... 111 8.2 Mathematical Expectation of Random

Variables ................................................................... 112 8.3 Statistical Meaning of Mathematical

Expectation ............................................................... 114 8.4 Main Properties of Mathematical Expectation ......... 116 8.5 Functions of Random Variables ............................... 117 8.6 Noncorrelated Random Variables ............................ 119 8.7 Variance of a Random Variable ............................... 120 8.8 Main Properties of Variance ..................................... 121 8.9 Other Characteristics of Dispersion ......................... 121 8.10 Moments of Random Variables of a Higher

Order ......................................................................... 122 8.11 Statistical Linearization ............................................. 123 8.12 Air-quality Comparison ............................................. 125 8.13 Questions and Exercises .......................................... 126

9. Numerical Characteristics of Random Variables: Quantiles ............................................... 129 9.1 Introduction ............................................................... 129 9.2 Probabilistic Meaning and Properties of

Quantiles ................................................................... 131 9.3 Statistical Meaning of Quantiles ............................... 135 9.4 Median, Quartiles, and Other Commonly Used

Quantiles ................................................................... 136 9.5 Application of Quantiles: Minimization of Mean

Losses Caused by Deviation of Random Variables from the Given Level ................................ 138

9.6 Evaluation of Time of Treatment .............................. 141

Contents xi


9.7 Planning of the Optimal Amount of Oxygen Supply ....................................................................... 141

9.8 Symmetrical Distribution ........................................... 142 9.9 Trace Metal Distribution ............................................ 143 9.10 Questions and Exercises .......................................... 144

10. Probability Distributions: Discrete Case .............. 145 10.1 Binomial (Bernoulli) Distribution ............................... 145 10.2 Numerical Characteristics of Binomial

Distribution ................................................................ 146 10.3 Multistage Processing System: Optimal Stage

Reserve Level ........................................................... 148 10.4 Hypergeometric Distribution ..................................... 149 10.5 Random Selection of Sample Sets from a

Dichotomous Collection ............................................ 152 10.6 Poisson Distribution .................................................. 154 10.7 Poisson Flow of Events ............................................ 157 10.8 Probabilities for the Number of Exceedances .......... 159 10.9 Probabilities of Major Floods .................................... 160 10.10 Questions and Exercises .......................................... 161

11. Probability Distributions: Continuous Case ......... 163 11.1 Introduction ............................................................... 163 11.2 Uniform Distribution .................................................. 163 11.3 Exponential Distribution ............................................ 165 11.4 Normal (Gaussian) Distribution ................................ 169 11.5 Properties of Normal Random Variables ................. 171 11.6 Application of Normal Distribution ............................ 173 11.7 Lognormal Distribution .............................................. 175 11.8 Application of Lognormal Distribution ....................... 177 11.9 Distribution of Solid Particles in Flowing

Water ........................................................................ 179

xii Contents


11.10 Mean Lifespan of a Bacterium ................................. 180 11.11 Occurrence of Strong Rainfall .................................. 180 11.12 Brownian Motion ....................................................... 181 11.13 Distribution of Grain Sizes ........................................ 181 11.14 Measurements of Trace Levels of Substances:

Normal-lognormal Distribution .................................. 182 11.15 Probabilistic Characterization of a Petroleum

Reservoir ................................................................... 183 11.16 Questions and Exercises .......................................... 188

12. Limit Theorems of the Probability Theory ............ 191 12.1 Introduction ............................................................... 191 12.2 Forms of Convergence for Random

Sequences ................................................................ 192 12.3 Chebyshevs Inequality ............................................. 192 12.4 Law of Large Numbers ............................................. 194 12.5 Central Limit Theorems ............................................ 197 12.6 Practical Use of Central Limit Theorems ................. 198 12.7 Application of Central Limit Theorems to

Bernoullis Scheme ................................................... 199 12.8 Application of Normal Distribution in Biological

Models ...................................................................... 199 12.9 Application of Chebyshevs Inequality ...................... 200 12.10 Maintenance of the Monitoring Stations ................... 202 12.11 Determination of the Number of Tests

Necessary for Confident Decision Making ............... 204 12.12 Questions and Exercises .......................................... 205

13. Probabilistic Decision Making ............................... 207 13.1 Introduction ............................................................... 207 13.2 Risk-assessment Methods ....................................... 208 13.3 Decision Making with Unknown Distributions .......... 210

Contents xiii


13.4 Decision Rules .......................................................... 211 13.5 Reconstruction of a Distribution Function

Based on a Subjective Assessment of Quantiles: Evaluation of the Available Amount of Groundwater Resources of an Aquifer ................. 214

13.6 Investigating Properties of Distribution Functions .................................................................. 218

13.7 Estimation of the Parameters of Distribution ............ 219 13.8 Properties of Good Estimators ................................. 219 13.9 Confidence Interval Construction ............................. 221 13.10 Testing of Hypotheses .............................................. 223 13.11 Air Pollution Investigation ......................................... 225 13.12 Questions and Exercises .......................................... 232

Appendices Appendix 1: Principles of Set Theory ................................. 235 Appendix 2: Methods of Counting ...................................... 245 Appendix 3: Statistical Comparison of an Original

Data Set with Its Subset in Oil Spill Studies ............. 249 Appendix 4: Standard Normal Distribution Function ......... 261

References ..................................................................... 263

Author Index .................................................................. 269

Index ............................................................................... 271

Front MatterPrefaceTable of Contents1. Introduction1.1 The Approach1.2 Overview1.3 Instructions

2. Experiments and Events2.1 Primary Notions2.2 Algebra of Events2.3 Relation of Implication2.4 Main Operations with Events2.5 Main Properties of the Operations with Events2.6 Theorem on the Decomposition of an Event into a Complete Set of Events2.7 Interpretation of Environmental Phenomena as Events of Experiments2.8 Questions and Exercises

3. Space of Elementary Events3.1 Preliminary Remarks3.2 Composition of the Space of Elementary Events3.3 Composition of the Space of Elementary Events for Air-quality Monitoring and Forecasting3.4 Characterization of the Eutrophication of a Bay Water3.5 Questions and Exercises

4. Probability of Random Events4.1 Random Events and Random Experiments4.2 The Concept of Probability of a Random Event4.3 Adequacy of Chosen Probabilistic Space to the Given Stochastic Experiment4.4 Corollaries of Probability Axioms4.5 Classic Definition of Probability4.6 Geometric Definition of Probability4.7 Statistical Definition of Probability4.8 Questions and Exercises

5. Conditional Probability and Stochastic Independence: Multistage Probabilistic Evaluation and Forecasting5.1 Conditional Probability5.2 Formula of Total Probability5.3 Bayes' Formula5.4 Examples of Application5.5 Independence of Events5.6 Multistage Probabilistic Assessment of Failure5.7 Simplified Probabilistic Model for Air-quality Forecasting5.8 Probability of a Water-purification System Being Functional5.9 Questions and Exercises

6. Bernoulli Distribution and Sequences of Independent Trials6.1 Bernoulli (Binomial) Distribution6.2 Sequence of Independent Trials and Its Mathematical Model6.3 Probabilistic Space for a Sequence of Independent Experiments6.4 Bernoulli Scheme of Independent Trials6.5 Examples of Application6.6 Application of the Bernoulli Scheme for Air-quality Assessment6.7 Questions and Exercises

7. Random Variables and Distribution Functions7.1 Quantities Depending on Random Events7.2 Mathematical Definition of a Random Variable7.3 Events Defined by Random Variables7.4 Independent Random Variables7.5 Distribution of a Random Variable: the Distribution Function7.6 General Properties of Distribution Functions7.7 Discrete Random Variables7.8 Continuous Random Variables7.9 General Properties of Distribution Density7.10 Distribution Function and Distribution Density of Functions of Random Variables7.11 Evaluating Probability of Soil and Groundwater Contamination7.12 Questions and Exercises

8. Numerical Characteristics of Random Variables: Mathematical Expectation, Variance, and Moments of Higher Order8.1 Introduction8.2 Mathematical Expectation of Random Variables8.3 Statistical Meaning of Mathematical Expectation8.4 Main Properties of Mathematical Expectation8.5 Functions of Random Variables8.6 Noncorrelated Random Variables8.7 Variance of a Random Variable8.8 Main Properties of Variance8.9 Other Characteristics of Dispersion8.10 Moments of Random Variables of a Higher Order8.11 Statistical Linearization8.12 Air-quality Comparison8.13 Questions and Exercises

9. Numerical Characteristics of Random Variables: Quantiles9.1 Introduction9.2 Probabilistic Meaning and Properties of Quantiles9.3 Statistical Meaning of Quantiles9.4 Median, Quartiles, and Other Commonly Used Quantiles9.5 Application of Quantiles: Minimization of Mean Losses Caused by Deviation of Random Variables from the Given Level9.6 Evaluation of Time of Treatment9.7 Planning of the Optimal Amount of Oxygen Supply9.8 Symmetrical Distribution9.9 Trace Metal Distribution9.10 Questions and Exercises

10. Probability Distributions: Discrete Case10.1 Binomial (Bernoulli) Distribution10.2 Numerical Characteristics of Binomial Distribution10.3 Multistage Processing System: Optimal Stage Reserve Level10.4 Hypergeometric Distribution10.5 Random Selection of Sample Sets from a Dichotomous Collection10.6 Poisson Distribution10.7 Poisson Flow of Events10.8 Probabilities for the Number of Exceedances10.9 Probabilities of Major Floods10.10 Questions and Exercises

11. Probability Distributions: Continuous Case11.1 Introduction11.2 Uniform Distribution11.3 Exponential Distribution11.4 Normal (Gaussian) Distribution11.5 Properties of Normal Random Variables11.6 Application of Normal Distribution11.7 Lognormal Distribution11.8 Application of Lognormal Distribution11.9 Distribution of Solid Particles in Flowing Water11.10 Mean Lifespan of a Bacterium11.11 Occurrence of Strong Rainfall11.12 Brownian Motion11.13 Distribution of Grain Sizes11.14 Measurements of Trace Levels of Substances: Normal-lognormal Distribution11.15 Probabilistic Characterization of a Petroleum Reservoir11.16 Questions and Exercises

12. Limit Theorems of the Probability Theory12.1 Introduction12.2 Forms of Convergence for Random Sequences12.3 Chebyshev's Inequality12.4 Law of Large Numbers12.5 Central Limit Theorems12.6 Practical Use of Central Limit Theorems12.7 Application of Central Limit Theorems to Bernoulli's Scheme12.8 Application of Normal Distribution in Biological Models12.9 Application of Chebyshev's Inequality12.10 Maintenance of the Monitoring Stations12.11 Determination of the Number of Tests Necessary for Confident Decision Making12.12 Questions and Exercises

13. Probabilistic Decision Making13.1 Introduction13.2 Risk-assessment Methods13.3 Decision Making with Unknown Distributions13.4 Decision Rules13.5 Reconstruction of a Distribution Function Based on a Subjective Assessment of Quantiles: Evaluation of the Available Amount of Groundwater Resources of an Aquifer13.6 Investigating Properties of Distribution Functions13.7 Estimation of the Parameters of Distribution13.8 Properties of Good Estimators13.9 Confidence Interval Construction13.10 Testing of Hypotheses13.11 Air Pollution Investigation13.12 Questions and Exercises

AppendicesAppendix 1: Principles of Set TheoryAppendix 2: Methods of CountingAppendix 3: Statistical Comparison of an Original Data Set with Its Subset in Oil Spill StudiesAppendix 4: Standard Normal Distribution Function

ReferencesAuthor IndexIndex

Probability in Petroleum and Environmental Engineering/11304_01.pdfCHAPTER 1

I N T R O D U C T I O N

Environmental issues have the highest possible priority in contem-porary science. There is no need to discuss the reasons for this situ-ation in detailthey are almost obvious. The purpose of appliedscience is to satisfy the demands of daily life, and today's realitiesare such that the deteriorating environment threatens the qualityof life in industrialized countries and people's existence in develop-ing nations.

THE APPROACHMany efforts by concerned scientists and engineers have beenmade in environmental health and safety, and to correct ecologicaldisturbances during the last 30 years, ever since the Rome's Club(Meadows et al., 1974) identified the main survival problem asadaptation to The Limits to Growth. Many useful, but many moreunsuccessful, environmental programs and projects were devel-oped and implemented. A new generation of progressive politi-cians was raised, who devoted their professional careers to thethemes of environmental protection and improvement. Owing tothe news media, general awareness of the educated public aboutpossible disasters has grown dramatically. Meanwhile, globally, theenvironmental problems are getting worse every day.

This does not mean, however, that the developed and imple-mented programs were useless. The tendency toward deteriorationof the environment is increasing because the rate of deterioration

is considerably higher than that of improvement. It is obvious thatone cannot count on rapid progress in the solution of environmen-tal problems. Only continuous, persistent, and meticulous workcan overcome the negative environmental tendencies.

The leading and organizing role in this work belongs to educa-tion. Many colleges and universities worldwide instituted academicprograms to prepare specialists in environmental protection. Al-most ail the programs, however, were highly specialized owing tothe prevailing scientific interests of a particular institution. Manydifferent environmental specialties have appeared, and there arehundreds of different titles of positions, occupants of which canhardly interact or even communicate effectively with each other.

To overcome these obstacles, leading environmental scientists bor-rowed from the industrial-military planning and development pro-cess and introduced the interdisciplinary concept of the systemsanalysis approach. This approach systematically uses mathematicalmodels of various forms for analysis and synthesis of suitable for-mal descriptions of a studied system together with their informalinterpretation. It is noteworthy that the form and complexity ofthe models developed depend on the purpose of their applicationand available resources that can be used for the solution of theproblem.

For environmental problems, the outcome of a monitored pro-cess frequently depends on numerous uncontrolled factors thathave a random nature. The results of monitoring the environmen-tal processes can be presented as outcomes of some stochastic ex-periments (real or conceptual). These outcomes can be conve-niently interpreted as corresponding random events, which occurin some stochastic experiments. This opens a wide area i>f applica-tions for the concepts and models based on probability theory inthe field of environmental issues.

In this book, the authors combined a rigorous and yet easyaxiomatic approach to probability theory with numerous examplesof environmental applications. The book is written as a treatise onbasic probabilistic concepts and methods. In each chapter theprobabilistic concepts are considered together with examples ofenvironmental applications. There is no doubt that such an ap-proach can provide students and practicing environmentalists with

a convenient; practical guide to the theoretical issues, and simulta-neously present specific conceptual approaches in the develop-ment of useful environmental applications.

For this book, the key word is chance, often associated with odds,occasion, event, accident, and luck. The situations in which one usesthese terms are endless in variety, but the common feature is thepresence of uncertainty. In playing roulette one puts a token on sayrouge without being sure of success. The general feeling about anynew business is that it is risky, and it is customary to evaluate thechances for success before starting it.

Evaluation (or estimation) of chances for occurrence of an eventof interest is even more important for environmental issues. Possibleatmospheric events, for example, are predicted with probabilisticestimates of their occurrence. Consequences of environmental proj-ects cannot even be expressed without the use of probabilistic terms.

In these situations, the events of interest may or may not occur.It is natural to call the events of this kind random. For randomevents, it is usually desirable to evaluate the chances of their ap-pearance (which requires some measure to express the chances). Itis customary to use probability of event as such a measure.

The computation or estimation of probabilities of the events ofinterest is the main focus of probability theory. In brief, probabilitytheory is a collection of methods for the evaluation of probabilitiesof the events of interest, based on given probabilities of some set ofprimary events. Conceptual basis of the theory was shaped duringthe last 70 years, and now it is possible to select its most fruitfuland applicable concepts and methods. Simultaneously, it is possibleto develop the most convenient pedagogic methods of presentingthe theory keeping in mind that the book is addressed to students,scientists, managers, and engineers specializing in environmentalissues.

Taking these circumstances into consideration, the authorshave chosen the direct way of presenting the main probabilisticconcepts, using examples of corresponding environmental issuesas illustrations and sources of probability problems. This book canbe regarded as a necessary introductory guide to probability theoryand its logic for the students and professionals who will have toameliorate and/or manage environmental messes. The authors

believe that numerous exercises and examples of environmentalapplications will convince the students and practicing environ-mentalists of the necessity and usefulness of the probabilisticmethodology.

OVERVIEWThe main goals and overview of the book are presented In Chapter1. The primary concepts of experiment and event are discussed at thebeginning of Chapter 2. The material that follows in Chapter 2 is,in considerable degree, traditional for the contemporary proba-bilistic courses and relates to the operations with events and varioussystems of events. Chapter 2 Is concluded with a specially developedscheme of Interpretation of an environmental phenomenon as aset of events of a particular stochastic experiment

An essential innovation Is introduced in Chapter 3. it relates tothe notion of an elementary eventf the basic theoretical concept thatallows constructing strict mathematical models for probabilisticcompositions. The elementary event Is strictly defined based on ageneral definition of event and the relation of implication. Notationc is used for this relation throughout the book. The expression AaBmeans: "If event A occurs, then event B occurs/' or "event A im-plies event B."

Any event of a particular experiment can be defined as a propercombination of elementary events. The space of elementary events isconsidered to be a union of all possible elementary events, it con-tains all possible outcomes of a specific stochastic experiment, inthe last section of Chapter 3, examples of composition of thespaces of elementary events for specific environmental problemsare considered in detail.

Chapter 4 covers the main probabilistic concepts, it Introducesthe idea of probabilistic space and presents various methods of itsconstruction and possible definitions of probability. The examplesof their application for description of environmental uncertaintiesare discussed at the end of Chapter 4.

The concept of conditional probability forms the core of Chapter5. Closely related issues, such as the Formula of Total Probability andBayes' Formula are discussed as the expansions of this concept. Inde-pendence of random events is one of the most applicable concepts for

the probability computation in composite probabilistic spaces. It isdefined and discussed in terms of conditional probability (if-thenpropositions).

Sequences of independent trials and related probability distribu-tions are discussed in Chapter 6. Before the calculation of any relatedprobabilities, one needs to compose a probabilistic space for thesequence of trials based upon the spaces of the individual trials. Theauthors explain in detail how one can construct a composite spacefor a multistage stochastic experiment Inasmuch as the Bernoulli frame-work of independent trials can be applied to numerous practical prob-lems, the writers also included the traditional material on Bernoulli(binomial) distribution and supporting theoretical information.

Random variables and their distribution functions are introducedin Chapter 7. Two numerical characteristics of distributions-moments and quantilesare considered in Chapters 8 and 9. Alltheoretical concepts are illustrated with applications addressingecological issues.

The most commonly used probability distributions for discreteand continuous random variables are described in Chapters 10 and 11.The normal distribution should be studied carefully, because it formsthe theoretical foundation of many applied mathematical models,and it has numerous applications in the evaluation of probabilitiesof events associated with random variables in the environmentalissues. The last section of Chapter 11 contains a detailed discussionand examples of the role of the lognormal distribution in modelingrandom variables characterizing concentrations of pollutants in var-ious media.

Chapter 12 contains general information on and examples ofthe probabilistic limit theorems of two kinds: the law of large numbersand central limit theorem. The authors discuss the conditions ofvalidity of these theorems and examples of their applications forstochastic processes in the environment.

Methods of decision making under uncertainty are discussed inChapter 13. They include techniques of assessing environmentalrisks and methods of estimating the properties of a random vari-able based on statistical sampling.

The book is designed for students, scientists, and engineers whohave completed a two-semester introductory calculus course. The

authors wish to assure the readers that with the basic background,the world of probability, with all its important applications in sci-ence as well as everyday life, is accessible to them. Given its mix ofabstract theory and practical issues, however, this book demandsconsiderable efforts on the part of the reader. It is important not tohurry but always to keep focus on the main concepts and the rela-tions among them. Readers are urged to work out the problemspresented at the end of each chapter. Their purpose is to highlightthe key concepts and to help readers to assimilate and apply theseconcepts before proceeding further. Moving forward gradually, thereader will be surprised, looking back, how much material has beenabsorbed.

The book is self-contained. It includes all necessary and auxil-iary information. In particular, students unfamiliar with elemen-tary concepts of set theory are referred to Appendix 1. Appendix 2contains necessary information on methods of counting. Thereshould, therefore, be little or no need to refer to other texts, exceptas sources of alternate viewpoints on the subject matter, or to digdeeper into specific problems. For this purpose, the authors recom-mend Frank Wolfs (1974) Elements of Probability and Statistics, andthe classical book entitled An Introduction to Probability Theory andIts Applications by William Feller (1968). The latter work can beused as a systematic reference. A recent development is the applica-tion of quantified logic and its associated visual information dia-grams to statistical analysis in decision making (Hammer, 1995;Adams, 1998).

During the last decade, several excellent books on applicationof probability theory and mathematical statistics were published(Devore, 2000; Kottegoda and Rosso, 1997; Millard and Neerchal,1999; Ott, 1995). These publications reflect a brisk pace of expan-sion of the probabilistic methodology in solving environmentalproblems.

Trying to simplify the theory and expedite its applications,many authors frequently provide a reader with ready recipes ofsolution of the probabilistic problems based on separate isolatedmethods and formulas. Important interrelationships among con-cepts and their potential applications that can be discovered onlyby systematic work on theoretical issues are mostly lost in this kind

of presentation. Meantime, the probabilistic methodology itselfprovides invaluable conceptual basis for mathematical modelingand analysis of a broad spectrum of environmental problems.

In this book, the authors consistently exploit the above ap-proach developing and reinforcing probabilistic knowledge andintuition of the reader to the level at which this knowledge can beused for construction of mathematical models of environmentalproblems of any nature. This feature distinguishes the book fromothers dealing with environmental issues.

INSTRUCTIONSThe material of the first 11 chapters could be considered manda-tory for an introductory course on the probability theory, and theauthors strongly recommend that instructors and students refrainfrom skipping any sections. The ideas presented in Chapters 12and 13 are important not only for their role in the mathematicalcontext in which they appear, but also for students' general educa-tion. This material cannot be covered rapidly, because it involvesabstract concepts that may be difficult to assimilate. If time is at apremium, the authors suggest stating the law of large numbers andcentral limit theorem in their simplest forms, with several examplesfrom Chapter 12.

Front MatterTable of Contents1. Introduction1.1 The Approach1.2 Overview1.3 Instructions

Index


E X P E R I M E N T S A N D E V E N T S

PRIMARY NOTIONSAny mathematical theory contains some primary undefinednotions that are explained in the examples of applications and arerelated to each other by some rules, for example, point, straight line,and plane in geometry. They cannot be reduced to the simpler con-cepts using definitions or relations of the theory. The main primaryconcepts of the probability theory are experiment and event.

Numerous examples of experiments can be easily found in natureand in all fields of human activity. Some of them are simple like theexperiments of elementary physics; others are complex, such asspecies breeding (genetics). In theory, one can also consider abstract(or conceptual) experiments, such as infinite tossing of a coin or infi-nite sampling of an environmental medium.

Researchers often design special experiments to investigate acertain property (or properties) of an object (or process) of interest.To conduct an experiment, one needs to ensure that certain condi-tions are met and perform prescribed actions. As a result of theseactions, one can observe some events. Based on occurrence of theseevents, the observer can make some inferences regarding the prop-erties of the object (or process) being investigated.

To specify the terms, let us consider several examples. The firstexample is taken from physics. Suppose an observer studies thelaws of free fall near the Earth's surface. The conditions of this

experiment are that several bodies are elevated above the Earth's sur-face and the possibility of allowing them to fall. Any actions thatforce the bodies to fall can be considered as the actions of experi-ment. Some of the events that can be observed in this experimentare: the time of falling of the first body is tt seconds, the time of fallingof the second body is t2 seconds, etc. Based on the observation ofthese events one can make some inferences regarding the laws offree fall.

Gambling provides a lot of good examples of stochastic experi-ments and random events (the words stochastic, probabilistic, andrandom are used in this book as synonyms). Suppose that anobserver (player) investigates the frequency of appearance of a cer-tain combination of points in roiling a set of dice, for example thecombination of appearance of two aces after rolling two dice. Sup-pose also that the observer decided to perform ten series of roilingwith 100 roils in each series. Assume that in these series two acesappeared three times in the first series, two aces appeared four times inthe second series, . . . , two aces appeared two times in the tenth series.Occurrence of these events provides the observer with the informa-tion regarding the relative frequency of occurrence of two aces in aseries of successive trials.

The third example describes a technological experiment. Supposethat some characteristic of an industrial product, density, for exam-ple, depends on the value of a certain variable factor, temperature, forexample. An experimenter fulfills the actions that consist in changingthe temperature. He can determine the density and measure the tem-perature. The pairs of these parameters provide the primary informa-tion, which characterizes the relationship between them.

The last example relates to an environmental problem. At theend of 1994, the South Coast Air-Quality Management District (LosAngeles, CA) announced that during that year, there were only 23days with bad air quality. There were 40 such days in 1992 and 127days in 1977. Based upon this information, the obvious conclusionhad been drawn that the air quality in Los Angeles had improvedconsiderably in 1994. This information had been obtained as aresult of a specially designed experiment. The experiment con-sisted of successive measurements of a set of parameters character-

izlng the concentrations of hazardous substances in the air at sev-eral points at the location of interest. If the chosen parameters didnot exceed assigned limits, then the air quality was good; other-wise, air was considered to be polluted.

It is not difficult to realize that an event of interest in this case isthe values of all state parameters of air quality were within the assignedlimits. Regular observations of these parameters allow monitoringenvironmental conditions and reveal the past and current tenden-cies in the changes of air quality over the area of interest.

After this short discussion, it is natural to call any set of actions ofinterest an experiment or trial As a result of an experiment, one canobserve some event (or events). In this book, it is considered thatcarrying out an experiment means fulfilling some prescribedactions under certain conditions. If an event appears regularlyevery time an experiment is performed, then it is called a determin-istic event Otherwise, an event is called random or stochastic.

It is noteworthy that one can discuss not only real experimentsand events but also imaginary (conceptual) ones. For example, it ispossible to consider an experiment that consists of a randomchoice of a point in a plane. It is clear that objects and actions ofthis experiment are imaginary. All events related to this experi-ment are also imaginary.

When it is necessary for theoretical considerations, one can pre-scribe some ideal features to real objects. For example, in a coin-tossing experiment one should consider that a coin's shape is strictlysymmetrical (a fair coin) in spite of the fact that this requirementcannot be completely satisfied in reality. Such idealization allows ex-cluding nonessential details from theoretical considerations.

In fact, any experiment considered in the probability theory isconceptual. Every time we have to define the conditions and possi-ble outcomes of the experiment. "When a coin is tossed, it does notnecessarily fall heads or tails; it can roll away or stand on its edge.Nevertheless, we shall agree to regard head and tail as the only possi-ble outcomes of the experiment. This convention simplifies the the-ory without affecting its applicability" (Feller, 1968, p. 7).

Any event, henceforth, will be called an event of an experiment oran observed event if the observer can conclude whether or not an

event occurred as a result of an experiment* The collection of all pos-sible events of an experiment is called a set of events of the experiment

The question of appropriate description of a set of events of agiven experiment is very important, because one needs suitablemeans to form the combinations of events. Returning to the exper-iment with air-quality monitoring, suppose, that for a particularregion, one needs only one parameter to characterize the air qual-ity, the concentration of carbon monoxide, for example, if oneintroduces the variable x for the concentration of carbon monox-ide at the location of interest, then the value of x can represent aparticular outcome of the considered experiment. Using the nota-tion X for a subset of real numbers, one can identify the set of ailevents of this experiment with the set of ail subsets, such as x X.It is noteworthy that the expression x e X is simply a short entryfor the event: variable x received a certain value from the set X after ful-filling the experiment

ALGEBRA OF EVENTSThe concept of an event is clarified and specified by suitable exam-pies of applications as well as by natural relations and reasonablerules on the operations with events imposed by the probabilitytheory. One can combine the events in various ways (they can beadded, subtracted, multiplied, etc). Applications of these opera-tions result in the new events of the considered experiment. Theseoperations differ from the operations of usual algebra. Many oftheir properties, however, are close to the properties of correspon-ding operations with real numbers; this justifies the use of thesame terms for them.

Sure eventDefinition 2.1. An event U is called the sure (or certain) event of agiven experiment if it occurs every time when the experiment iscarried out

Suppose, for example, that h is the height of a man chosen atrandom. Then, the event (h > 0) is sure. In the experiment with cointossing, the event (heads or tails) is sure. For any experiment one canpoint out some events that are sure events of this experiment.

Impossible eventDefinition 2.2. An event V is called impossible for a given ex-periment if it never occurs in the experiment.

For example, the event (heads and tails) cannot appear in a one-coin-tossing experiment. It is obvious that one can always pointout an impossible event (or events) for any experiment. It is note-worthy that the concepts of sure and impossible events make sensefor a given experiment only. Thus, tails and heads, for instance, canappear jointly in the experiment by tossing two coins.RELATION OF IMPLICATIONThere are situations when an event B is necessarily implied byanother event A. In such cases, A and B are linked by the relation ofimplication. This important concept can be defined as follows.

Definition 2.3. Let A and B be events of the same experiment.The event A implies B (written down as A c B) if B appears in agiven experiment on occurrence of A.

Consider, for example, the experiment with the rolling of fair die:event A = (two appeared) and event B = (an even number appeared). It isobvious that A c B (event B is implied by event A).

Remark 2.1. If A c B, where A and B are some events of a givenexperiment, then B can occur without A. IiA occurs, however, thenB appears without fail.

Remark 2.2. Sometimes A c B, because A is the cause of B, but itis not necessary. Consider, for example, event B = (coin landed) inthe coin-tossing experiment. It is clear that A c B for any event A ofthis experiment.

Remark 2.3. If >4 cB, then B can appear (1) later than A, (2) simul-taneously with A, or (3) earlier than A.

It is interesting to describe an example when B happens earlierthan A. Suppose there are two handguns: loaded and unloaded. Anexperimenter selects one of them at random and shoots at a target.Consider the following events:

P = (the loaded handgun was chosen),Q = (the unloaded handgun was chosen), andS = (bullet's mark appeared on the target).

It is obvious that S a P; however, P occurred before 5.

Main properties of the relationof implicationLet us introduce a set of all observed events of a given experimentand use the notation for it. Suppose that A, B, C are some eventsfrom O, U is a sure event, and ^Is an impossible event. Then

1. AaA.2. A c B ^ c C ^ A c C .3. A c t / (sure event is implied by any event of experiment).4. V cz A (impossible event implies any event of experiment).Property 1 is obvious. Let us verify property 2. If B is implied by

A and C is implied by B, then, according to the definition, if A hap-pens, B must also occur, and therefore, C occurs. Hence, if A happens,then C happens without fail.

Consider property 3. If any event A happens, then U must occurwhen the experiment is carried out. Property 4 can be justified onthe basis of consideration that inasmuch as V never happens, thenproperty 4 states nothing about the occurrence of A. A statementthat asserts nothing can always be accepted as formally true. Onecan always interpret property 4 in this way.

Relation of equalityDefinition 2.4. An event A is equal to an event B (A = B) if A c Band B a A.

It is noteworthy that equality A-B means that the events A andB occur jointly, but it does not mean that they are identical. Tounderstand this fact, it is useful to consider the following example.Suppose that some marksman hits his target without fail. Event Ais the fact of shooting, whereas event B is the appearance of a markon the target. Afterward, A = B; however, it is clear that A and B arenot the same.

The concept of transitive property is expressed in the followingform:

(A = B), (B = C) =* (A = C).This is the main property of the relation of equality that is often usedfor the comparison of events. In the previous example, suppose that

the target is changed every time when the mark appears on it. Con-sider the event C = (target has changed). Then A = B and B = C. Thetransitive property implies A = C, which is obvious in this case.

MAIN OPERATIONS WITH EVENTSlike numbers, events can be added, subtracted, and multiplied. Asa result of these operations, one can obtain new events with fea-tures that are determined by the combined events and appliedoperations. Although the meaning of these operations is different,they have many common properties with the corresponding opera-tions from usual algebra that justifies the use of the same name forthem. One can also use the corresponding terms from the set the-ory that in many cases can be more convenient, because, as onewill see later, an event can be identified with a certain set.

Sum (union) of eventsDefinition 2.5. Let A, B e , where O is the set of all events of agiven experiment. An event, representing the occurrence of A or B(or both), is called the sum (or union) of the events A and B. Stan-dard notation for this operation is A + B or A u B.

It is clear that the statement "event A + B appeared in the exper-iment" is equivalent to the statement "either A or B happened inthe experiment" (possibly both of them). Owing to this circum-stance, one can say that A or B occurred in the experiment.

The operation can be generalized for an arbitrary collection ofevents (A i e I), where I is some set of indices. Namely, the eventi^E A (r ui. Then the difference A\B (or A - B) ofevents A and B is the event that represents occurrence of A but not B.

For example, if A = (even number came out) and B = (two faced up)in the die-rolling experiment, then A\B = (four or six faced up).

The operations discussed above (sum, product, etc.) are called thealgebraic operations with events. They allow constructing compoundevents with desirable properties on the basis of some primary events.

MAIN PROPERTIES OF THE OPERATIONSWITH EVENTSFormal expressions and algebraic interpretation of the main prop-erties are presented in Table 2.1.

It is noteworthy that properties l-3(a) in Table 2.1 are similar tothe properties of sum and product for real numbers. Let us proveproperties 2(b), 3(a), and 4(b). The others are implied either by thedefinitions or can be justified by similar considerations.

Property 2(b). (AB)C occurred (AB) and C occurred A andB and C occurred => A and (B and C) occurred 04+Ee) .3. (A e T) => Ac e .It is customary in probability theory to suppose that a primary

collection of events of any experiment forms an algebra. It is anatural convention that allows using the algebra of events as amathematical model of an experiment. Property 2, for example,means that, if an observer is aware of occurrence of some event A

and another event B, then he can conclude whether A + B occurredin the experiment. Properties 1 and 3 can be interpreted in a simi-Lar way.

In advanced mathematical literature (e.g., Feller, 1971), it isassumed that a primary system of events W forms so-called sigtna-algebra (^-algebra). This is a system of events with properties 1 and3, which satisfies the following condition:

4. (A1 % Aze%...,Ane 4>, ...) => L / = 1 2 . A1 e *F.

INTERPRETATION OF ENVIRONMENTAL PHENOMENAAS EVENTS OF EXPERIMENTSProbability theory broadens considerably the concept of an experi-ment. This concept was formed primarily in physics as a means totest theoretical hypotheses regarding a certain studied phenome-non. For example, on investigating the laws of free fall above theEarth's surface, the investigator assumes that the time of descentfrom a given height to the Earth's surface depends on the mass of afalling body. This hypothesis can be tested by carrying out anexperiment in which several bodies with different masses are used.

An important condition of any physical experiment is that thefactors influencing the outcome of an experiment are kept (orremain) the same except one, the effect of which is being studied.For the above example, the only variable factor is the mass of afalling body; height, air density, and initial velocity of the bodyshould be constant.

In physics, the factors that influence the outcome of an experi-ment are named variables. A variable that can be intentionallychanged is called the manipulated (or control) variable. If one cancontrol all the variables of an experiment, then such an experi-ment is called the controlled experiment The outcome of an experi-ment is sometimes called the responding variable (or response). Theinterpretation of the results of a series of experiments provides evi-dence to either support or reject the hypotheses on properties ofthe responding variable.

For environmental phenomena in nature, one cannot (as a rule)design a special experiment for testing the hypotheses of interest,because it is simply impossible to stabilize ail essential variables on

the prescribed levels. In these situations, the attention is focusedexclusively on the observation of a studied process. The influenceof multiple input factors is considered to be negligible if inferencesabout the properties of process are made on the basis of numerousobservations. The only source of objective information in this caseis an evaluation of some numerical characteristics of realizations ofthe process of interest.

Sometimes the same characteristics can be used for quite differ-ent purposes. Consider, for example, the data from Table 2.2 thatdescribes the dynamics of world population and the averagemethane gas concentration in the atmosphere.

These data that characterize environmental investigation can beused for different purposes. Methane is a gas that does not supportlife. Therefore, analyzing the dynamics of the methane concentra-tion, one can conclude that general world air quality deterioratedconsiderably during the last 50 years.

On the other hand, if one considers the main sources of meth-ane in the Earth's atmosphere, then it is possible to make some

Table 2.2. World population and average methane concentration inatmosphereYears Methane Concentration, ppm World Population, billion2000 1.80 6.11990 1.70 5.31980 1.50 4.21970 1.40 3.51960 1.30 3.01950 1.25 2.51940 1.15 2.01900 1.00 1.51850 0.85 1.21800 0.75 1.01750 0.74 0.81700 0.72 0.71650 0.70 0.61600 0.70 0.5

general conclusions regarding the level of civilization's development.The sources of methane gas include: (1) cattle's digestion, (2) ricepaddies, (3) decomposition of organic matter (in absence of oxygen),(4) natural gas leakage (from underground natural gas and oildeposits to the Earth's surface), (5) oil/natural gas production, and(6) termite metabolism. The volumes of methane entering the at-mosphere from these sources can characterize the intensity of differ-ent aspects of human activity and, consequently, provide someimplicit information about the level of civilization's development.

The data in Table 2.2. are a typical example of outcome of an en-vironmental investigation. It is noteworthy to emphasize that thedata do not have an absolute meaning: their interpretation dependson the purposes of conducting a particular experiment. The samelevel of uncertainty in description of the process of interest andinterpretation of the results of observation is common for stochasticexperiments. There is a natural and convenient correspondencebetween the environmental observations and stochastic experi-ments that allows applying the framework, terminology, and math-ematical formalism of a stochastic experiment to the environmentalprocess analysis. This correspondence is presented in Table 2.3.

This correspondence can be used as a guideline for descriptionand interpretation of any environmental process of interest as acertain stochastic experiment. The advantages of this possibilityare obvious: after appropriate interpretation one can apply allmeans of the rigorous, well-developed theory of probability for de-scription and analysis of the environmental processes.

Table 2.3. Correspondence between environmental processes and stochasticexperimentsEnvironmental ProcessEnvironmental outcomeCombination of outcomes"Mathematical model"Frequency of occurrence of out-come in a series of experiments

Stochastic ExperimentStochastic eventOperations with eventsAtgebra of eventsProbability of the event of interest

For the discussed example, the stochastic experiment consists ofthe successive methane concentration observations. The stochasticevent of interest is that the methane concentration belongs to thechosen interval of real numbers. Different events can be consideredas belonging to the methane concentration to different intervals ofreal numbers. Thus, the algebra of events can be introduced imme-diately as the algebra of intervals (sets of points). For these inter-vals, the probability measure, which allows characterizing the cor-responding frequencies of occurrence for the environmentalprocess of interest, can be introduced. After that, the process ofinterest (methane concentration) can be investigated using theprobability theory.

QUESTIONS AND EXERCISES1. What is called an experiment? Give several examples of

experiments.2. Explain the difference between real and conceptual experiments.3. What is called an observed event? Describe the class of

observed events in the experiment of air-quality monitoring.4. Give the definitions of sure and impossible events. Do they

have an absolute meaning?5. Give the definition of the relation of implication for events

of the same experiment. Does relation AaB mean that A is acause of Bl Should B appear later than A in the experiment?

6. State and verify the main properties of the relation ofimplication.

7. What does the relation of equality mean? What is the mainproperty of the relation of equality?

8. Give the definition of sum of two events. Generalize it forthe sum of more than two events. What system of events iscalled the complete set of events?

9. Define the product of two events. Give the general definitionof product for more than two events. Which events are calledmutually exclusive?

10. Verify the formula A\B = ABC.

11. State properties 1-4 of the operations with events. Verify ailof them. Generalize property 3 for arbitrary collection ofevents. Prove it.

12. Verify the following relations:a. A + A = A t ACZ = Ab. AA=A g. A\B**A\ABc. A + V = A h. A + B = A + (B\A)d. AV=V I A+B = A + (B\AB)e. A + U=U j . if AcB, thenB = A+ (\A)

13. State the theorem on decomposition of an event into a com-plete set of events. Prove the theorem.

14. Define the algebra of events. Why is it assumed that a set ofobserved events must possess the structure of algebra? Whatset of events ts called the E-algebra of events?

15. What analogy exists between the concepts of environmentalprocess and stochastic experiment? How can one use thisanalogy for description and analysis of environmentalprocesses?

Front MatterTable of Contents2. Experiments and Events2.1 Primary Notions2.2 Algebra of Events2.3 Relation of Implication2.4 Main Operations with Events2.5 Main Properties of the Operations with Events2.6 Theorem on the Decomposition of an Event into a Complete Set of Events2.7 Interpretation of Environmental Phenomena as Events of Experiments2.8 Questions and Exercises

Index


S P A C E O F E L E M E N T A R Y E V E N T S

PRELIMINARY REMARKSDiscussion of the relations among events revealed that any event Aof an experiment is implied by at least two events: A itself and animpossible event (V). In a general case, other events implying Acan be found. Consider, for example, the die-rolling experiment(Chapter 2). Here, the event A = (even number appeared) is impliedby any of the following five events: A, V, (two), (four), and (six)occurred in this experiment.

There are situations, however, when a certain event A is impliedby A and V only. Such an event is further called elementary event.The following definition specifies this important concept.

Definition 3.1. An event co * V is called elementary if there areonly two events that imply it in the experiment: co itself and V.

An event that is neither elementary nor V is known as a com-pound event

Remark 3.1. Whether a particular event co is elementary orcompound depends, of course, on a primary system of events ({>.Consider, for instance, the die-rolling experiment. Introduce thefollowing events: A = (even number), B = (odd number), Co1 = (one), co2= (two), . . . ^ (O6 = (six). At first, assume an observer cannot see themarks on the die's face but receives information from anotherobserver. Another observer lets the first observer know whether anodd or even number of dots appears, but does not specify its value.

That is a reason for choosing the four events A, B, U, and V as theset of observed events for the first observer. One needs to includethe events U and V in order to obtain the algebra of events. Undersuch a choice of a primary system of events, the events A and B areelementary. The second observer, however, would definitely decideto include the events O1, o 2 / . . . , O6 into the set of observed eventsof this experiment Then, the event A, for example, is not elemen-tary with respect to this wider set of observed events. Keeping thisin mind, it is reasonable sometimes to consider an elementaryevent with respect to a primary system of events $

As an example, let us discuss again the die-roiling experimentassuming that the experimenter observes the number of dots thatappears on the die's face. The same notations O1, o2, . . . , O6 areused for the observed events. One can describe now the set of ailobserved events for this experiment.

Let \|/ be the notation for collection of all subsets of the set (1, 2 , . . . ,6), Le., each element of y is a subset of the set of numbers (1, 2 , . . . ,6). It can be, for example, number (1), the pair (2, 5), empty set, etc.Consider now the family of all events of the kind (j e G), where / isthe number appeared and Gey. Denote this family as . It is not dif-ficult to verify that 0 and P(B) > 0. How could one find the probabilityP(B I A) under these conditions?

Using the definition of conditional probability and Eq. 5.3, oneobtains

P(B\ A)= P(AB)IP(A),and

P(B U) = P(A \B)P(B)/P(A). (5.6)Equation 5.6 is known as Bayes' formula. To generalize it, one canreplace B by H. e (H1, . . ., HN) and rewrite A in Eq. 5.6 in accor-dance with the formula of total probability. Thus, one obtains

P(Hf U) = P(A \H)P(H)lfrf(H^P(A lH.). (5.7)The last expression is also called Bayes' formula.

EXAMPLES OF APPLICATIONExample 53. Oysters are grown on three marine farms for the pur-pose of pearl production. The first farm yields 20% of total produc-tion of oysters, the second yields 30%, and the third yields 50%.The share of the oyster shells containing pearls in the first farm is5%, the second farm is 2%, and the third farm is 1%. What is theprobability of event A that a randomly chosen shell contains apearl?

Solution: Introduce the events:

H1 = (shell was grown on the first farm),H2 = (shell was grown on the second farm), andH3 = (shell was grown on the third farm).It is easy to understand that these three events form a complete

set of events. Using the above notations and the given conditions,one obtains

P(H1) = 0.2, P(H2) =0.3, P(H3) = 0.5, P(^ l | H1) = 0.05,P(A IH2) = 0.02, P(A IH3) = o.oi.

According to Eq. 5.5,

P(A)=IP(H)P(A]H,)= 0.2 x 0.05 + 0.3 x 0.02 + 0.5 x 0.01 = 0.021.

Example 5,4. Under the conditions presented above, a ran-domly chosen shell contains the pearl. What is the probability ofevent that this shell was grown at the first farm?

Solution: In the notations of example 5.3 it is necessary to findP(H1 U). According to Eq. 5,6,

P(H11 A) = P(A I H1)P(H1)IP(A) = (0.05 x 0.2)/(0.021) - 0.48.Example 5.5. Quality of water in a riverThe two indices BOD (biochemical oxygen demand) and DO

(dissolved oxygen) are among the parameters that determine thequality of water in a river. BOD is a relative measure of the biologi-cally degradabie organic matter present in the water. The higherthe DO level, the better self-cleaning ability of the water. For the"healthy" river, the BOD has to be low and DO has to be high.

To investigate the quality of water in the Blackwater River inEngland under study, concurrent measurements of BOD and DOwere conducted at 38 stations along the river's flow (Kottegoda andRosso, 1997, p. 706, Table E. 1.3). As a rule, there is a strong correla-tion between the BOD and DO: the higher the BOD, the lower theDO, and vice versa. This correlation is presented by two segmentsof the best fit line in Fig. 5.1. The line y = f(x) divides the area ofpossible values of the two parameters into two zones: zone S withthe "high ability to self-clean" (below the line) and zone [/with the"low ability to self-clean" (above the line).

For the discussed problem one can introduce the following fullgroup of mutually exclusive events (Kottegoda and Rosso, 1997):

H1 = (water is unpolluted and unhealthy),H2 = (water is polluted and unhealthy),H3 = (water is polluted and healthy), andH4 = (water is unpolluted and healthy).

Based on the data presented in Table E. 1.3 in Kottegoda and Rosso(1997, p. 706), one needs to evaluate the probability that water inthe river is unpolluted and healthy provided that it has a high abil-ity for self-cleaning. In formal notations, one needs to find theconditional probability P(HJS).

Using information from Fig. 5.1, one can formalize the eventsof interest in the following way:

Figure 5.1 An illustration of the problem of evaluation of the probability of"blooming."

H1 = (DO < 7.5 n BOD < 3.2),H2 = (DO < 7.5 o BOD > 3.2),H3 = (DO > 7.5 n BOD > 3.2), andH4 = (DO > 7.5 n BOD < 3.2).

Identifying probabilities of these events with their relative frequen-cies, one obtains

P(H1) = 2/38 = 0.05, P(H2) = 17/38 = 0.45,P(H3) = 0/38 = 0, P(H4) = 19/38 = 0.5.Conditional probabilities are defined by corresponding relative

frequencies:

P(SfH1) = 1.00, P(SIH2) = 11/38 = 0.29,P(SIH3) = 0, P(SIH4) = 19/38 = 0.50.

According to the formula of total probability (Eq. 5.5),P(S) = P(SIH1)P(H1) + P(SIH2)P(H2) + P(SIH3)P(H3) + P(SIH4)P(H4).

On substitution of the corresponding numbers,

P(S) = 1.00 x 0.05 + 0.29 x 0.45 + 0x0 + 0.50 x 0.50= 0.43.

DO, mglL

BOD,

m

g/L

Application of Bayes' formula yields

P(HJS) = P(SZH4)P(H4)IP(S)and

P(H4IS) = 0.50 x 0.50/0.43 = 0.25/0.43 = 0.58.

INDEPENDENCE OF EVENTSLet A and B be two events of the same experiment and P(B) > 0.Event A does not depend (stochastically) on event B if

P(AlB)=P(B).It is noteworthy that if A does not depend on B and P(A) > 0, then Bdoes not depend on A, because, according to Bayes' formula,

P(B I A) = P(A I B)P(B)IP(A). (5.8)Using independence of A and B, one obtains

P(A)P(B)IP(A)^P(B).Thus, one can formulate the following definition.

Definition 5.2. Two events A and B are called (stochastically)independent if one of them does not depend on another or hasprobability of zero. The second part of the definition is, of course, aconvenient complement.

Two events A and B are independent of each other if and only if

P(AB) = P(A)P(B). (5.9) Assume that A and B are independent. If P(B) > 0, thenP(AB) = P(A \ B)P(B) = P(A)P(B).

If P(B) = 0, then0 0. (5.12)This gives an explicit formula for the probability of interest.Remark 5.2. Applicability of the modelEquation 5.12 can be employed to evaluate the applicability of

the considered model for long-term forecasting. Tables 5.1 and 5.2contain the results of calculation Pn for n = 2 ("short-term predic-tion") and n = 5 ('long-term prediction") for various p using Eq.5.12. The data in these tables indicate that one should not expectgood results for long-term forecasting by applying the consideredmodel

Table 5.1 Probability of good air quality on nth day (n = 2).p 0.5 0.6 0.7 0.8 0.9Pn 0.5 0.52 0.58 0.68 0.82

Table 5.2. Probability of good air quality on nth day (n = 5).p 0.5 0.6 0.7 0.8 0.9Pn 0.5 0.5002 0.5051 0.5390 0.6640

PROBABILITY OF A WATER-PURIFICATION SYSTEMBEING FUNCTIONALOne of the most fruitful concepts for the calculation of probabili-ties of compound events is the concept of independence. To illus-trate its application, one can consider the following problem.

A certain water-purification system contains five filters. Eachone of the five filters of the water-purification system functionsindependently with probability of 0.95. The purification system isconsidered to be safe if at least two filters function properly. Findthe probability of the purification system being safe.

Let F. be the event that the /th filter functions, and S be theevent that the purification system is safe. It is easier to calculate theprobability of the complementary event Sc. To understand it, oneneeds to express the events S and Sc in terms of the event F1.. For theevent Sc, one has

Sc = [(F1Y n (F2Y n (F3Y n (F4Y n (F5Y] u[(F1) n (F2Y n (F3Y n (F,Y n (FSY] u[(F1Y n (F2) n (F3)' n (F4Y n (F5)'] u[(F1)' n (F2Y n (F3) n (F4)' n (F5)'] u[(F1)' n (F2)' n (F3)' n (F4) n (F5)'] u[(F1Yn (F2Yn(F3Yn (F4)' n(Fs)]

All the events of the last expression unified by the signs uare mutually exclusive. Therefore, the probability of event S' can becalculated as the sum of probabilities of the above components.Taking into consideration the fact that all filters have the sameprobability of functioning properly, one obtains

P(SO = P[(F,Y n (F2Y n (F3)' n (F4)' n (F5)']+ 5Pf(F1) n (F2)' n (F3)' n (F4)' n (F5)']

P(SO = (1 - 0.95)5 + Sx 0.95 x (1 - 0.95)4 0.3 x 10"4P(S) = 1-P(S') = 0.9997.

QUESTIONS AND EXERCISES1. Give the definition of conditional probability of an event A

for a given event B. What are the probabilistic and statisticalmeanings of this definition?

2. Calculate P(B | B), P(Q \ B), and P(0 i B). Compute P(A I B) if (a)AB = 0, (b) AcB, and (c) BcA.

3. How can one calculate the probability of event AB if P(A \ B)and P(B) are given?

4. State and prove the formula of total probability.5. State and prove Bayes' formula.6. Give the definition of independence for two events. What are

the probabilistic and statistical meanings of this definition?Consider examples of events that are independent (from yourpoint of view).

7. Give examples of pairwise and mutual independence ofevents. Try to clarify whether or not pairwise independenceimplies mutual independence.

Front MatterTable of Contents5. Conditional Probability and Stochastic Independence: Multistage Probabilistic Evaluation and Forecasting5.1 Conditional Probability5.2 Formula of Total Probability5.3 Bayes' Formula5.4 Examples of Application5.5 Independence of Events5.6 Multistage Probabilistic Assessment of Failure5.7 Simplified Probabilistic Model for Air-quality Forecasting5.8 Probability of a Water-purification System Being Functional5.9 Questions and Exercises

Index


B E R N O U L L I D I S T R I B U T I O N

A N D S E Q U E N C E S O F

I N D E P E N D E N T T R I A L S

BERNOULLI (BINOMIAL) DISTRIBUTIONDefinition 6.1. The distribution of probabilities is a collection ofevents that forms a complete set together with the probabilities ofthese events.

Assume that Av A2, . . . , An are independent in a collection ofevents and each one of them has the same probability of occurrencep. Let Bm(O t0). During time tt - tQ, the particle is displaced by thedistance Ix1 - xo| depending on the direction of movement of theparticle. The trajectory of the particle is determined by the impactsof molecules of the liquid surrounding the particle. Moments oftime and magnitude of impacts are random in this experimentConsequently, the variable Ix1-X0I is random also.

Example 7.4. The lifespan of a certain individualFor an individual, life longevity is a random variable.Example 7.5. Bacteria in a microscope sightThe number of bacteria in a microscope sight is a random variable.

MATHEMATICAL DEFINITION OF A RANDOM VARIABLETo define a random variable, one needs to introduce at first a spaceof elementary events for a stochastic experiment associated withthe variable. For examples 7.1 and 7.4, the space can be the set ofall individuals from which selection is made. For example 7.2, it isthe collection of all sample sets of a given size, which can be takenfrom a tested lot. For example 7.3, it is the space of all continuousvector-functions with the domain [t0, t j and values of functions inthe three-dimensional Euclidian space. For example 7.5, it can bethe limited set of whole numbers.

Definition 7.1. Informal definition of a random variableLet Q be the space of elementary events of some stochastic experi-ment. Any function defined on Q and taking numerical values(and, possibly, values -H*>, -)> and T| = ti(co). A particular value of a ran-dom variable corresponding to a certain elementary event is oftencalled the realization of a random variable.

EVENTS DEFINED BY RANDOM VARIABLESSuppose that X is a subset of the set of all real numbers R. With anyrandom variable , one can associate the event that a realization of belongs to X. This fact can be written briefly as ( e X). As a rule,a semibounded interval (-, x) or a bounded interval [JiC1, X2) is cho-sen as X, where x, xv X2 Bite some given real numbers. In the casewhere X = (-, oo) and #(->) = a,

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Probability in Petroleum and Environmental Engineering