6.5 The (a, b, 0) class
The following definition characterizes the members of this class of distributions.
Definition 6.4 Let pk be the pf of a discrete random variable. It is a member of the (a, b, 0) class of distributions provided that there exists constants a and b such that
This recursion describes the relative size of successive probabilities in the counting distribution. The probability at zero, p0, can be obtained from the recursive formula because the probabilities must sum to 1. The (a, b, 0) class of distributions is a two-parameter class, the two parameters being a and b. By substituting in the probability function for each of the Poisson, binomial, and negative binomial distributions on the left-hand side of the recursion, it can be seen that each of these three distributions satisfies the recursion and that the values of a and b are as given in Table 6.1. In addition, the table gives the value of p0, the starting value for the recursion. The geometric distribution, the one-parameter special case (r = 1) of the negative binomial distribution, is also in the table.
It can be shown (see Panjer and Willmot [89, Chapter 6]) that these are the only possible distributions satisfying this recursive formula.
The recursive formula can ...