Sir, I am stuck on this question

Question 8 Observations X = {X1,. . . , Xn} are independent identically distributed, being Gauslan with N(p.,d'2), where p. is unknown while 0'2 is known. Suppose that p. itself (i.e. the prior distribution) is a Gaussian random variable with mean Y and variance 32. We would like to nd and analyse the MAP estimate of 1.1. 1. Let C = fxbc}. Using BayesI rule, show that the posterior probability can be written as N _ l I 4126911 1 _l 2 2 fuleIIXJC (Emits 1 )(ne ) 2. Show that N X it l-l _ ll "V 2 _ 2 i=1 0' B by maximizing the log posterior. 3. By rearranging the terms, show that 11 = [32 21:1 in: + 02? BIN + (II 4. What happens when N grows large? Note that we have I N UML=gxt which is the maximum likelihood estimate of p