Handbook of Probability

Chapter Nine

Moment-Generating Function

9.1 Introduction/Purpose of the Chapter

We chose to introduce the generating function and the moment-generating function after the chapter on characteristic function. This is contrary to the traditional and historical approach. In modern-day probability the characteristic functions are much more widespread simply because they exist and can be constructed for any random variable or vector regardless of its distribution. On the other hand, generating functions are defined only for positive discrete random variables. The moment-generating function does not exist for many discrete or continuous random variables. Despite this fact, when these functions exist, they are much easier to work with than the characteristic functions (recall that the characteristic function takes values in the complex plain). Thus, we feel that the knowledge of these functions cannot miss from the culture of anyone applying probability concepts.

9.2 Vignette/Historical Notes

According to Roy (2011), Euler found a generating function for the Bernoulli numbers around 1730's. That is, he defined numbers B_n such that

Around the same time, DeMoivre independently introduced the method of generating functions in a general way (similar to our definition). He used this method to encode all binomial probabilities in a simple polynomial function. Laplace (1812) applied the Laplace ...