Using a long rod that has length , you are going to lay out a square plot in which the length of each side is u. Thus the area of the plot will be u. However, you do not know the value of , so you decide to make n independent measure- ments X, X...., X, of the length. Assume that each X, has mean (unbiased measurements) and variance o.

a. Show that is not an unbiased estimator for u. [Hint: For any rv Y, E(Y) (Y)+ [E(Y). Apply this with Y = X.]

b. For what value of k is the estimator X - kS unbiased for u? [Hint: Compute E(X - kS).]