Let X1,X2, ...,Xn be a sample from a populatioanollowing a Poisson distribution with parameterp > 0: \"k P(X = k) = e'\"F,k = 0,1,2, Recall that E(X) = Var(X) = p and let)? be the sample mean. This question aims at the estimation of the parameter p = P(X = 0) = e'\". Define two estimators of p as 11. 1 p1 = EZKXJ- = 0) andpz = e X. i=1 (a) Show that :31 is a consistent estimator of p. [4] (b) Show that 131 is asymptotically normal and give the mean and the variance of this limiting distribution. [4] (c) Show that 132 is a consistent estimator of p. [4] (d) Show that 1'52 is the maximum likelihood estimator of p. [4] (e) Determine the asymptotic distribution of 132. [4] (f) Which of the two estimators would you recommend? Give your reasons. Hint. Forx > 0,x 1 2 lnx