A population evolves in generations. Let $Z_{n}$ be the number of members in the $n$th generation, with $Z_{0}=1$. Each member of the $n$th generation gives birth to a family, possibly empty, of members of the $(n+1)$ th generation; the size of this family is a random variable and we assume that the family sizes of all individuals form a collection of independent identically distributed random variables each with generating function $G$.

Let $G_{n}$ be the generating function of $Z_{n}$. State and prove a formula for $G_{n}$ in terms of $G$. Determine the mean of $Z_{n}$ in terms of the mean of $Z_{1}$.

Suppose that $Z_{1}$ has a Poisson distribution with mean $\lambda$. Find an expression for $x_{n+1}$ in terms of $x_{n}$, where $x_{n}=\mathbb{P}\left\{Z_{n}=0\right\}$ is the probability that the population becomes extinct by the $n$th generation.