2 GENERATING FUNCTIONS
A generating function packs a whole distribution into a single function of an auxiliary variable. Nothing is lost in the packing — the distribution can always be read back out — and what is gained is that operations which are awkward on distributions become easy on functions. The one that matters most is addition: the distribution of a sum of independent variables is a convolution, which is a genuinely difficult integral or sum, but the generating function of the sum is simply the product of the generating functions. That single fact carries most of the next three sections.
Three versions are in use, and they differ only in what is substituted for the auxiliary variable.
2.2 Properties of the Probability Generating Function
2.3 Sums and Linear Functions
2.4 The Moment Generating Function
2.5 The Cumulant Generating Function
2.6 Uniqueness and Continuity
2.7 Practice Problems
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