In this problem we will work with the Beta-Binomial setup from class: (see below for the rest of the question)

In this problem we will work with the Beta-Binomial setup from class: H w Beta(o, ,8) X | H w Binomialn, 5'] Recall that Eta) = . and em?) = gm from the mean a variance of the Beta distribution. The idea is that we cannot see 9, but we want to estimate it based on observed data given by X. Assume that ea, {3 are known constants. Our goal in this problem is to estimate 6' as accurately as possible. [a] Suppose that we pick a xed value t as our \"guess" for 9. Calculate the mean squared error [MEI-3) of this [non-random} estimator, Elli? - tll, in terms of i. What value of i will minimize the MSE? What is the MSE at this value of t? How does your answer relate to the mean andj'or variance of 9? [b] Now let's instead use X {a as our estimator. Let's calculate the MSE, E({X;'n 9F). Hint: use the tower law (aka. law of total expectation). [c] For what values of n, o, ,3 does the rst estimator {just guessing the value t, for the optimal value of t] have lower or higher MSE than the second estimator (given by X fa)