Generating Functions

Abstract

It is a fundamental principle of mathematics to map a class of objects that are of interest into a class of objects where computations are easier. This map can be one to one, as with linear maps and matrices, or it may map only some properties uniquely, as with matrices and determinants.

In probability theory, in the second category fall quantities such as the median, mean and variance of random variables. In the first category, we have characteristic functions, Laplace transforms and probability generating functions. These are useful mostly because addition of independent random variables leads to multiplication of the transforms. Before we introduce characteristic functions (and Laplace transforms) later in the book, we want to illustrate the basic idea with probability generating functions that are designed for $\mathbb{N} _{0}$-valued random variables.

In the first section, we give the basic definitions and derive simple properties. The next two sections are devoted to two applications: The Poisson approximation theorem and a simple investigation of Galton–Watson branching processes.