Solve

Exercise A-4 Let Xi, X2,..., Xn be an iid random sample where Xi ~ Normal (y, 07), 4 unknown, and 07 unknown. In Discussion, two similar exercises were explored: finding the MLE for 4 when o2 was known, and finding the MLE for o? when yu was known. This time, both are unknown, but those Discussion exercises will still be helpful. (a) Find the MLE's for both pw and o%. Hint: The likelihood and log-likelihood functions for both will be the same. But when you get to Step 3, you'll get two expressions: one from taking the derivative with respect to p, and the other by taking the derivative with respect to o2. (b) Show that the MLE 6? is a biased estimator for o2 when p is unknown. Calculate the value of the bias of . (c) Based on your answer to part (b), what would you need to do to the MLE 6 to create an unbiased estimator for o? when yp is unknown? Show that your new statistic based on the MLE 6 is unbiased. Hint: What is the name/symbol of that \"new statistic'? You have seen it before