Generating Functions

This chapter introduces various generating functions encountered in statistics. Generating functions find a variety of applications in engineering and applied sciences. There are four popular generating functions used in statistics: probability generating function (PGF), moment generating function (MGF), cumulant generating function (CGF), and characteristic function. As the PGF of a random variable generates probabilities, it can be used to generate the sum of left tail probabilities (CDF). The MGF of a random variable is used to generate the moments algebraically. The characteristic function uniquely determines a distribution. The CGF is slightly easier to work with for exponential, normal, and Poisson distributions. There are two types of factorial moments known as falling factorial and raising factorial moments. The classical probability generating function of discrete distributions is extended to get a new generating function for the CDF. This is then used to derive the mean deviation by extracting just one coefficient.