Let X,..., Xn be independent Binomial (m, 0) random variables, where m is known and [0, 1] is the statistical parameter. Let x,...,n be modelled as realizations of X,..., Xn. (a) (5 marks) Let p(0) be the chosen prior for and p(x) the derived posterior. Define the Bayes estimate of 0, 0, and state how the Bayes estimate is associated with the posterior distribution for each of the loss functions: L(0,0) = (0-0), L(0,0) = 10-01 and L(0,0) = I(0 + 0). (b) Let the prior distribution for be Beta(a, 3) where a and 8 are nonnegative constants. i. (5 marks) Show that the posterior distribution for is also a Beta distribution and find its parameters. ii. (3 marks) Derive expressions for the posterior mean and the posterior variance for 0. iii. (5 marks) Consider the posterior mean as an estimator of 0. Show that the sequence of estimators, corresponding to increasing values of n, is consistent. (c) Let the prior distribution for be discrete with p(0) = - [ 0 {}, }}, 0 otherwise. i. (3 marks) Derive the posterior distribution for 0. ii. (4 marks) Show that the Bayes estimate associated with the loss function L(0,0) = I(0 + 0) is 0 = if and only if i