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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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