ABSTRACT
This chapter is devoted to moment generating functions. A moment generating function is an alternate representation of its corresponding distribution function. It is obtained as a transformation of the distribution, and it is an encoded form for all of the information contained in its corresponding distribution. The key information provided by the moment generating function are the distribution moments. The chapter starts with an explanation of how the moment generating function is constructed as an expectation and shows that moment generating functions can be constructed for distributions on both discrete and continuous random variables. It is also shown that empirical distributions as well as standard distribution families have moment generating functions. It is explained and shown that distribution moments are obtained as derivatives of the moment generating functions. It is then further shown how to use moment generating functions to obtain probability distributions on sums of independent random variables. Those sums are usually called convolutions. The chapter continues by developing the extension of moment generating functions to joint distributions, marginal distributions and conditional distributions. Here again, both discrete and continuous cases are treated.
