Let $X$ be a random variable taking values in the non-negative integers, and let $G$ be the probability generating function of $X$. Assuming $G$ is everywhere finite, show that

$$G^{\prime}(1)=\mu \text { and } G^{\prime \prime}(1)=\sigma^{2}+\mu^{2}-\mu$$

where $\mu$ is the mean of $X$ and $\sigma^{2}$ is its variance. [You may interchange differentiation and expectation without justification.]

Consider a branching process where individuals produce independent random numbers of offspring with the same distribution as $X$. Let $X_{n}$ be the number of individuals in the $n$-th generation, and let $G_{n}$ be the probability generating function of $X_{n}$. Explain carefully why

$$G_{n+1}(t)=G_{n}(G(t))$$

Assuming $X_{0}=1$, compute the mean of $X_{n}$. Show that

$$\operatorname{Var}\left(X_{n}\right)=\sigma^{2} \frac{\mu^{n-1}\left(\mu^{n}-1\right)}{\mu-1}$$

Suppose $\mathbb{P}(X=0)=3 / 7$ and $\mathbb{P}(X=3)=4 / 7$. Compute the probability that the population will eventually become extinct. You may use standard results on branching processes as long as they are clearly stated.