Let X ~ Bin(n, p). Consider the estimator (a) Find Elp ], and hence state the bias be?(p?). [3 marks] (b) By considering the moment generating function for a binomial distribution Mx(t) = (1-p+pet)", show that: . EX = np . EX? = n(n - 1)p- + np . EX3 = n(n - 1)(n -2)p3 + 3n(n - 1)p3 + np . E[X] =n(n-1)(n -2)(n -3)p# + 6n(n - 1)(n -2)p3 + 7n(n - 1)p2 + np [8 marks] (c) Hence, or otherwise, calculate MSEp: (p?). [4 marks] (d) Show that [2 marks] E P(1 - P) p(1 - p) n - 1 n (e) Using the Parts (a) and (d), find an unbiased estimator for p-. [1 mark] (f) BONUS: Calculate the MSE for your estimator in Part (e) [5 marks]. [Question total: 18]