Paper 2, Section II, F

Probability | Part IA, 2017

For a positive integer N,p[0,1]N, p \in[0,1], and k{0,1,,N}k \in\{0,1, \ldots, N\}, let

pk(N,p)=(Nk)pk(1p)Nkp_{k}(N, p)=\left(\begin{array}{c} N \\ k \end{array}\right) p^{k}(1-p)^{N-k}

(a) For fixed NN and pp, show that pk(N,p)p_{k}(N, p) is a probability mass function on {0,1,,N}\{0,1, \ldots, N\} and that the corresponding probability distribution has mean NpN p and variance Np(1p)N p(1-p).

(b) Let λ>0\lambda>0. Show that, for any k{0,1,2,}k \in\{0,1,2, \ldots\},

limNpk(N,λ/N)=eλλkk!\lim _{N \rightarrow \infty} p_{k}(N, \lambda / N)=\frac{e^{-\lambda} \lambda^{k}}{k !}

Show that the right-hand side of ()(*) is a probability mass function on {0,1,2,}\{0,1,2, \ldots\}.

(c) Let p(0,1)p \in(0,1) and let a,bRa, b \in \mathbb{R} with a<ba<b. For all NN, find integers ka(N)k_{a}(N) and kb(N)k_{b}(N) such that

k=ka(N)kb(N)pk(N,p)12πabe12x2dx as N\sum_{k=k_{a}(N)}^{k_{b}(N)} p_{k}(N, p) \rightarrow \frac{1}{\sqrt{2 \pi}} \int_{a}^{b} e^{-\frac{1}{2} x^{2}} d x \quad \text { as } N \rightarrow \infty

[You may use the Central Limit Theorem.]

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