# 7.1 Compound frequency distributions

A larger class of distributions can be created by the processes of compounding any two discrete distributions. The term *compounding* reflects the idea that the pgf of the new distribution *P*_{S}(*z*) is written as

where *P*_{N}(*z*) and *P*_{M}(*Z*) are called the *primary* and *secondary* distributions, respectively.

The compound distributions arise naturally as follows. Let *N* be a counting random variable with pgf *P*_{N}(*z*). Let *M*_{1}, *M*_{2}, … be i.i.d. counting random variables with pgf *P*_{M}(*Z*). Assuming that the *M*_{j}s do not depend on *N*, the pgf of the random sum *S = M*_{1} + *M*_{2} + ··· + *M*_{N} (where *N* = 0 implies that *S* = 0) is *P*_{S}(*z*) = *P*_{N}[*P*_{M}(*z*)]. This is shown as

In insurance contexts, this distribution can arise naturally. If *N* represents the number of accidents arising in a portfolio of risks and {*M*_{k} : *k* = 1, 2, …, *N*} represents the number of claims (injuries, number of cars, etc.) from the accidents, then *S* represents the total number of claims from the portfolio. This kind of interpretation is not necessary to justify the use of a compound distribution. If a compound distribution fits data well, that may be enough justification itself. Also, there are other motivations for these distributions, as presented in Section 7.18.

### EXAMPLE 7.1

Demonstrate that any zero-modified distribution ...

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