Suppose that a population evolves in generations. Let $Z_{n}$ be the number of members in the $n$-th generation and $Z_{0} \equiv 1$. 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 with the same generating function $G$.

Let $G_{n}$ be the generating function of $Z_{n}$. State and prove a formula for $G_{n}$ in terms of $G$. Use this to compute the variance of $Z_{n}$.

Now consider the total number of individuals in the first $n$ generations; this number is a random variable and we write $H_{n}$ for its generating function. Find a formula that expresses $H_{n+1}(s)$ in terms of $H_{n}(s), G(s)$ and $s$.