Lecture 7:Generating Functions

andThe Laplace Transform

Department of Electrical EngineeringPrinceton University

October 4, 2013

ELE 525: Random Processes in Information Systems

Hisashi Kobayashi

Textbook: Hisashi Kobayashi, Brian L. Mark and William Turin, Probability, Random Processes and Statistical Analysis (Cambridge University Press, 2012)

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9.1 Generating function For a given sequence {fk; ; k = 0, ±1, ±2, …}, its generating function is defined by

In systems analysis, the name “Z-transform” has gained wide acceptance, where

9.1.1 Probability-generating function (PGF) For given probabilities {pk ; k =0, 1, 2, …} where

its probability generating function (PGF) is defined by

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The number q-1 is called the radius of convergence.

9.1.1.1 Generating function of the complementary distribution

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The region |z|<q -1 is called the region of convergence of the function (9.6).

3

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9.1.1.2 Expectation and factorial moments

If we let z →1 in (9.10), we obtain P’ (1)= Q(1). Thus, the expectation is

Differentiate (9.11) once more, and using the relation P”(z)=2Q’(z)+(z-1)Q”(z)

The last expression is called the nth factorial moment.

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9.1.2 Sum of independent variables and convolutions

called the convolution summation or simply convolution.

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9.1.3 Sum of a random number of random variables

Let {qn } be the probability distribution of N , and PN(z) be its PGF:

We are interested in the probability distribution {rS} of SN.

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9.1.4 Inverse transform of generating functions

Taylor series expansion of P(z):

Cauchy’s residue formula:

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9.1.4.1 Partial-fraction expansion method

Assume that D(z)=0 has d distinct roots.

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When the roots are not all distinct, i.e., the ith root has multiplicity mI .

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By extending the above method, we find

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9.1.4.2 Asymptotic formula in partial-fraction expansion

Assume that z1 is the smallest root in absolute value among all the d distinct roots,

Consider the case m1 =2. Recall (9.67)

The term

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Consider the general case m1 ≥ 1.

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9.1.4.3 Recursion method

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9.2 Laplace transform method 9.2.1 Laplace transform and moment generationLet X be a nonnegative and continuous RV.

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9.2.2 Inverse Laplace transform

where c > α.

Remark: For a nonnegative RV , the LT of its PDF f(t) always exists, because

9.2.2.1 Partial-fraction expansion method

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If n = d,

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Asymptotic formula in partial-fraction expression

c.f. Section 9.1.4.2 (pp. 224)

We must find the root whose real part is smallest in absolute.Let - λ1 be such a smallest root with multiplicity m1.

In Example 9.9

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9.2.2.2 Numerical-inversion methodRecall (9.95):

Setc > α

where δ= T/N and

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