# 6.5 The (*a, b*, 0) class

The following definition characterizes the members of this class of distributions.

**Definition 6.4** *Let p _{k} 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, *p*_{0}, 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 *p*_{0}, 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 ...

Get *Loss Models: From Data to Decisions, 4th Edition* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.