Question Two Suppose that X1, . .., Xn are i.i.d. N(0, #) with density function f(x; 0) = VO =e TER, 020 V27T where 0 is an unknown parameter. a [2 marks] Find the maximum likelihood estimator for 0. O Omle = O Omle = = n I'M O Omle = 1 O n Omle = En , X? O Omle = X2b) [3 marks] A prior T(@) on 0 is assumed to be Gamma(3, =) with the density of a Gamma(a, B) given by: 1 T( ) =T(Q ) Ba a, B > 0. Find the posterior distribution h (0| X). O 0 | X ~ Gamma 2 + 2, Zil X? + 4 O 0| X ~ Beta 2 + 3, ELIX? +4, O 0 | X ~ Gamma + 2. LiX/ +4 2 O 0 | X ~ Gamma + 3, 2 Et- X? + 4 O 0 | X ~ Gamma +3, Zin X? + 4 2[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) xx 0 exp X Hint: The mean of a Gamma(a, B) distribution is aB. O Obayes = 10X O Obayes = 9X O Ohayes 10 O Obayes = O Ohayes = Xd) [2 marks] Using a zero-one loss, determine the form of the test y when testing the hypothesis Ho : 0 1 in the Bayesian context. if P(0 > 1 X) 1 X ) > 0.5, O 1 if P(0 0.5, O 1 if P(0 > 1 X) > 0.5 40* = 10 if P(0 0.5 * =10 if P(0 > 1 X) 1