(a) Explain what is meant by the term 'branching process'.

(b) Let $X_{n}$ be the size of the $n$th generation of a branching process in which each family size has probability generating function $G$, and assume that $X_{0}=1$. Show that the probability generating function $G_{n}$ of $X_{n}$ satisfies $G_{n+1}(s)=G_{n}(G(s))$ for $n \geq 1$.

(c) Show that $G(s)=1-\alpha(1-s)^{\beta}$ is the probability generating function of a non-negative integer-valued random variable when $\alpha, \beta \in(0,1)$, and find $G_{n}$ explicitly when $G$ is thus given.

(d) Find the probability that $X_{n}=0$, and show that it converges as $n \rightarrow \infty$ to $1-\alpha^{1 /(1-\beta)}$. Explain carefully why this implies that the probability of ultimate extinction equals $1-\alpha^{1 /(1-\beta)}$.