Let X1, X2, . . . , Xn be a random sample from the normal distribution with mean and variance 2, i.e., N (, 2). (1) Show that the maximum likelihood estimator (MLE) of is the sample mean X = n n1 X i . i=1 (2) Show that the variance of the MLE is 2/n. (3) Consider another estimator Y = (X1 + X2)/2. Show that Y is an unbiased estimator. (4) From the class, we learned that the sample means are unbiased, so MLE for this problem is unbiased as well. Briefly explain why the MLE is better than the estimator Y in part (3), even if both estimators are unbiased