 Solution

A Poisson distribution with parameter $\mu$ has probability generating function $G_N(s)=e^{\mu (s-1)}$. The preceding part gives $G_F(s)=e^{\mu p(s-1)}$, the probability generating function of a Poisson distribution with parameter $\mu p$. To prove independence, retain both counts:Consequently the joint probability generating function factors asComparing the coefficients of $s^f{t}^k$ proves that the joint probabilities are the products of the marginal probabilities. Hence $F$ and $S$ are independent, with distributions $\mathrm{Poisson}(\mu p)$ and $\mathrm{Poisson}(\mu (1-p))$. This is Poisson thinning, including the degenerate counts when $p=0$ or $p=1$.