# 7.2 Further properties of the compound Poisson class

Of central importance within the class of compound frequency models is the class of compound Poisson frequency distributions. Physical motivation for this model arises from the fact that the Poisson distribution is often a good model to describe the number of claim-causing accidents, and the number of claims from an accident is often itself a random variable. There are numerous convenient mathematical properties enjoyed by the compound Poisson class. In particular, those involving recursive evaluation of the probabilities were also discussed in Section 7.1. In addition, there is a close connection between the compound Poisson distributions and the mixed Poisson frequency distributions that is discussed in more detail in Section 7.3.2. Here we consider some other properties of these distributions. The compound Poisson pgf may be expressed as

(7.9) where Q(z) is the pgf of the secondary distribution.

### EXAMPLE 7.7

Obtain the pgf for the Poisson–ETNB distribution and show that it looks like the pgf of a Poisson–negative binomial distribution.

The ETNB distribution has pgf for β > 0, r > − 1, and r ≠ 0. Then the Poisson–ETNB distribution ...

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