Transcription of Chapter 4: Generating Functions - Auckland
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Chapter 4: Generating Functions - Auckland
74 Chapter 4: Generating FunctionsThis Chapter looks at Probability Generating Functions (PGFs) fordiscreterandom variables. PGFs are useful tools for dealing with sums and limits ofrandom variables. For some stochastic processes, they alsohave a special rolein telling us whether a process willeverreach a particular the end of this Chapter , you should be able to: find the sum of Geometric, Binomial, and Exponential series ; know the definition of the PGF, and use it to calculate the mean, variance,and probabilities; calculate the PGF for Geometric, Binomial, and Poisson distributions; calculate the PGF for a randomly stopped sum; calculate the PGF for first reaching times in the random walk; use the PGF to determine whether a process willeverreach a given sums1. Geometric Series1 +r+r2+r3+..= Xx=0rx=11 r,when|r|< formula proves thatP x=0P(X=x) = 1 whenX Geometric(p):P(X=x) =p(1 p)x Xx=0P(X=x) = Xx=0p(1 p)x=p Xx=0(1 p)x=p1 (1 p)(because|1 p|<1)= Binomial TheoremFor anyp, q R, and integern,(p+q)n=nXx=0 nx pxqn that nx =n!
The probability generating function gets its name because the power series can be expanded and differentiated to reveal the individual probabilities. Thus, given only the PGFGX(s) = E(sX), we can recover all probabilitiesP(X = x). For shorthand, write px = P(X = x). Then GX(s) = E(sX) = X∞ x=0 pxs x = p 0+ p1s + p2s 2+p 3s 3+ p 4s 4+ ...
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