variance

10. Using a long rod that has length m, you are going to lay out a square plot in which the length of each side is m. Thus the area of the plot will be m2. However, you do not know the value of m, so you decide to make n independent measurements X1, X2,… Xn of the length. Assume that each Xi has mean m (unbi- ased measurements) and variance s2.
a. Show that X 2 is not an unbiased estimator for m2. (Hint: For any rv Y, E(Y 2) = V(Y) + [E(Y )]2. Ap- ply this with Y = X.)
b. For what value of k is the estimator X 2 – kS 2 un- biased for m2? [Hint: Compute E(X 2 – kS 2).]