Question Two Suppose that X1, . . . ,X are i.i.d. Poisson[3) with probability mass function 695" :13! f[m;]=Pg{X=m}-= :_ mE{D,1,2,...}, i9>0, where 6' > U is an unknown parameter. a} [2 marks] The maximum likelihood estimator for 6'2 is given by X2. One can Show that 13(22): 9 +92. 1'1 Which of the following would he a bias-corrected MLE for I92. 2(1 X}2 X2 n}_{ E +2?2 11 b3 [3 marks] A prior 11:9] on 3 is assumed to be Gammax, 13) with density function 1 Hum\" Find the posterior distribution h{3|X} where 3 = 2:1 Xi. 7(3) : 3'14 67:, or,,8 > I]. ]X~Gamma(s,n;u) C! H | X~Gamma(,3+s,n+) H | XwGamma(a-i,n+3) S Submit () [2 marks] Find the Bayesian estimator with respect to the loss function L(0, a) = (0 - a) if the posterior distribution was h(0|X) o 0* exp ( X + 4 ) Hint: The mean of a Gamma(a, B) distribution is aB. Obayes = 5(X + 4) O Obayes = 4(X + 4) O 5 O'bayes = (X + 4) O Obayes = (X + 4) 5 O Obayes = 20Xd) [2 marks] Using a zero-one loss, determine the form of the test y when testing the hypothesis Ho : 0 2 2 versus H1 : 0 2 X) 2 X) 2 0.5, O 40* = 10 1 if P(0 0.5, O if P(0 > 2 X) > 0.5 * o if P(0 0.5 if P(0 > 2 X) 2