Generating Functions in Elementary Probability Theory

Abstract

For many years certain authors of textbooks in and statistics have used the term without first defining generating As in Wilks (1, p. 114) the usual definition of of a random variable X (with moments of all orders) is E(e8X) where stands for expected value. Also as in Wilks (1, p. 114) the same author may define probability of an integral-valued random variable X as E(sx) without letting the reader in on the fact that he is using the term generating in two different ways. Worse yet, as in Wilks (1, p. 114) the author may refer to E(sx) as the moment Although E(sx) can be and should be used to find factorial moments, it does not generate them in either of the two ways in which the previously mentioned functions generate their respective sequences. In this paper we are interested in three sequences: (1) the sequence of probabilities { pi, i = 0, 1, 2, ... where pi = P(X = i), i.e. X is an integral valued random variable; (2) the sequence of moments { E (Xi) , i = 0, 1, 2, ... I}; and (3) the sequence of factorial moments {E(X[iJ), i = 0,1,2, .... For a sequence of real numbers {ai, i = 0, 1, 2, . . .} there are two kinds of functions in common use as in Riordan (2, p. 19): the ordinary function and the exponential function. We define these functions for the given sequences.