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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.

a sum using the traditional probability function. The PGF transforms a sum into a product and enables it to be handled much more easily. Sums of random variables are particularly important in the study of stochastic processes, because many …

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