# 9.4 Analytic results

For most choices of distributions of *N* and the *X*_{j}s, the compound distributional values can only be obtained numerically. Subsequent sections in this chapter are devoted to such numerical procedures.

However, for certain combinations of choices, simple analytic results are available, thus reducing the computational problems considerably.

### EXAMPLE 9.7

(*Compound negative binomial–exponential*) Determine the distribution of *S* when the frequency distribution is negative binomial with an integer value for the parameter *r* and the severity distribution is exponential.

The mgf of *S* is

With a bit of algebra, this can be rewritten as

which is of the form

where

the pgf of the binomial distribution with parameters *r* and β/(1 + β), and *M**_{X}(*z*) is the mgf of the exponential distribution with mean θ(1 + β).

This transformation reduces the computation of the distribution function to the finite sum, that is,

When *r* = 1, *S* has a compound geometric distribution, and in this case the ...

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