Let X : (X1, . . . , Xn)T be a collection of Nd. random variables with probability density function (pdf): f(.L, )w3, U is an unknown parameter. Let x : (:51, . . . , LE\")T be a random sample observed from this distribution (this is a realisation of X). (a) Find a onedimensional sufcient statistic T(X) for 6. [2] (b) Calculate the Maximum Likelihood Estimator of 9. [3] (c) Is the Maximum Likelihood Estimator of 9 unbiased for 9? Justify your answer. [10] A Bayesian statistician assumes that the prior distribution for 6 is given by 9 N \"[0, a], where a E R+, (1 7E 0. (d) Determine the posterior distribution of 6 based on the sample X. [15] Suppose that the following sample of X1, . . . , Xn is obtained, where n : 8: 25.6, 16.4, 11.8, 10.8, 12.2, 10.4, 16.8, 15.0 (e) Assuming that the prior distribution for 19 is such that 9 N M [0, 10], determine the Bayes' estimate of 6 where the loss function is given by L(9,d) = (d (9)2, where d E R