Question 1 [16 marks] Let X and Y be independent random variables with X ~ Binomial(n, p), Y ~ Binomial(2n, p). The following expressions provide two possible estimators of p: 2n PI = AX +LY and p2 = X+Y. 3n From now on, you may assume that Binomial(m, p) has expectation mp and variance mp(1 - P). (a) Demonstrate that E(X?) = np(1 - p+ np) and E(Y?) =2np(1 -p + 2np). (b) Explain why both pi and p2 are unbiased for p. (c) Show that var(PI) = p(1 - p) p(1 - p) 00 1 0O and var(p2) = 091 - 72. n. (d) Derive the efficiency of the estimator p, relative to p2. (e) Show that both p, and p2 are consistent estimators of p. (f) Which estimator would you prefer? Why? (g) Find approximate normal distributions for X and Y when n is large. (h) Find approximate normal distribution for pi and p2 when n is large