Paper 2, Section II, F

Probability | Part IA, 2018

(a) Consider a Galton-Watson process (Xn)\left(X_{n}\right). Prove that the extinction probability qq is the smallest non-negative solution of the equation q=F(q)q=F(q) where F(t)=E(tX1)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

P(X1=1)=1/4,P(X1=3)=3/4\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 nNn \in \mathbb{N}, let YnY_{n} be a random variable with distribution Poisson(n)\operatorname{Poisson}(n). Show that

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

in distribution, where ZZ is a standard normal random variable.

Deduce that

limnenk=0nnkk!=12\lim _{n \rightarrow \infty} e^{-n} \sum_{k=0}^{n} \frac{n^{k}}{k !}=\frac{1}{2}

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