Generating Functions

Abstract A generating function is a way of documenting a sequence of items, such as probabilities or moments, using generators of formulas for the individual terms in a sequence. Common generating functions have the form of power series or of exponential series. For discrete distributions, they are particularly valuable for combining sequences of probabilities in various ways. This article discusses moment generating functions and gives examples. It then discusses the cumulant generating function, which is sometimes simpler to handle, and gives examples. Probability generating functions and factorial moment generating function are useful tools for studying discrete distributions.