Question 2 Let X1, X2, ..., Xn be a sample from a population X following a Poisson distribution with parameter u > 0: P ( X = k) = e-Muk KI , k = 0,1,2, ... Recall that E(X) = Var(X) = u and let X be the sample mean. This question aims at the estimation of the parameter p = P(X = 0) = e-M. Define two estimators of p as P1 = = > I(x, = 0) and p2 = e-x. (a) Show that p, is a consistent estimator of p. [4] (b) Show that p1 is asymptotically normal and give the mean and the variance of this limiting distribution. [4] (c) Show that p2 is a consistent estimator of p. [4] (d) Show that p2 is the maximum likelihood estimator of p. [4] (e) Determine the asymptotic distribution of pz. [4] (f) Which of the two estimators would you recommend? Give your reasons. Hint. For x > 0, x - 1 2 Inx. [4]