7.1 Poisson: .
Negative 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)