(a) Consider a Galton-Watson process $\left(X_{n}\right)$. Prove that the extinction probability $q$ is the smallest non-negative solution of the equation $q=F(q)$ where $F(t)=\mathbb{E}\left(t^{X_{1}}\right)$. [You should prove any properties of Galton-Watson processes that you use.]

In the case of a Galton-Watson process with

$$\mathbb{P}\left(X_{1}=1\right)=1 / 4, \quad \mathbb{P}\left(X_{1}=3\right)=3 / 4$$

find the mean population size and compute the extinction probability.

(b) For each $n \in \mathbb{N}$, let $Y_{n}$ be a random variable with distribution $\operatorname{Poisson}(n)$. Show that

$$\frac{Y_{n}-n}{\sqrt{n}} \rightarrow Z$$

in distribution, where $Z$ is a standard normal random variable.

Deduce that

$$\lim _{n \rightarrow \infty} e^{-n} \sum_{k=0}^{n} \frac{n^{k}}{k !}=\frac{1}{2}$$