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# CHAPTER 7 SOLUTIONS

### 7.1   SECTION 7.1

7.1 Poisson: .

Negative binomial: .

Binomial: .

7.2 The geometric–geometric distribution has pgf

The Bernoulli–geometric distribution has pgf

The ZM geometric distribution has pgf

In PBG(z), replace 1 − q with (1 + β1)−1 and replace β with β2(1 + β1) to see that it matches PGG(z). It is clear that the new parameters will stay within the allowed ranges.

In PGG(z), replace q with (1 − p0)(1 + β*)/β* and replace β with β*. Some algebra leads to PZMG(z).

7.3 The binomial–geometric distribution has pgf

The negative binomial–geometric distribution has pgf

In the binomial–geometric pgf, replace m with r, β(1 − q) with β2, and β with (1 + β1)

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