In addition to the moment generating function (MGF) and characteristic function (CF) methods discussed in the preceding chapter, there are two other related methods that are frequently used in the study of probability theory. They are the generating function and the Laplace transform (LT).

Discrete RVs often assume integers or integral multiples of some unit, as is the case in counting applications and discrete-time systems. Then, the generating function method will be found to be a convenient device in probability analysis. When a random variable is continuous but nonnegative (e.g., waiting time and service time in a queueing system), we can make use of the rich theory of LTs in the analysis.

Since the CF exists for all distribution functions, both discrete and continuous, why should we study all these other transform methods that seem redundant? Certainly the CF should suffice in most situations, but generating functions and LTs are preferred whenever they are applicable, partly because their notation is somewhat simpler than that of the CF, and partly because there is a rich theory behind the generating function and LT methods, both of which have been widely used as operational methods in system theory that involves differential and integral equations. Thus, it is important for us to be sufficiently familiar with these transform methods to study the literature on probability theory and its applications.

Generating function

The notion of generating function can be more general than the probability generating function (PGF) that we will primarily discuss in this section.