Problem 4. Let X1, ..., Xn ~ N(B, 62), and suppose that the prior distri- bution on 0 is N(u, T2). (Here we assume that o', u and T2 are all known. (1) Prove that the posterior distribution of 0 is also normal, with mean and variance given by E(0|Xn)= Xn+ 82 72 + 02 / n 02 / n+ 72 1 , 02+2 Var(0|Xn) = 02 + NT 2 . (2) How will the posterior distribution look like when n - oo? (3) Intuitively, as the prior information becomes more vague, the poste- rior mean as a Bayes estimator for 0 should rely more on the sample information. Examine E(0|Xn) as 2 - oo, and explain why your result is consistent with the intuition