Question

Suppose that Xi,.. Xn are independent and identically distributed Bernoulli random variables, with mass function P (Xi = 1) = p and P (Xi = 0) = 1-p for some p (0,1) (a) For each fixed p є (0,1), apply the central limit theorem to obtain the asymptotic distribution of Σ.Xi, after appropriate centering and normalisation. (b) Derive the moment generating function of a Poisson(A) distribution. (c) Now suppose that the probability p in the mass function of the sequence {Xi,., X) decreases to 0 as the sample size n increases: p-λ/n for some λ > 0, Show that, as n → oo, Sn , Xi converges in distribution to a Poisson (A) random variable. How does this result compare to the limit distribution obtained in part (a)?

Can someone help me with part (c), (with detailed explanation)

Answer 1

First WKT if Xi's are IID then SUM(Xi) follow binomial distribution with parameters n and p. now we obtaon limiting case as-therefore the distribution of sum(Xi) vconverges to a poisson distribution with parameter lemda. here we can say that for p is less then 0.10, with large n we obtain poisson distribution. and another thing is that sum(Xi) follow poisson distribution for large n and small p but mean of x follow S.N distribution for large n.

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