Probability | Part IA, 2005

Describe the Poisson distribution characterised by parameter λ>0\lambda>0. Calculate the mean and variance of this distribution in terms of λ\lambda.

Show that the sum of nn independent random variables, each having the Poisson distribution with λ=1\lambda=1, has a Poisson distribution with λ=n\lambda=n.

Use the central limit theorem to prove that

en(1+n1!+n22!++nnn!)1/2 as ne^{-n}\left(1+\frac{n}{1 !}+\frac{n^{2}}{2 !}+\ldots+\frac{n^{n}}{n !}\right) \rightarrow 1 / 2 \quad \text { as } \quad n \rightarrow \infty

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