Using a long rod that has length , you are going to lay out a square plot in which the length...

Using a long rod that has length , you are going to lay out

a square plot in which the length of each side is . Thus the

area of the plot will be 2. However, you do not know the

value of , so you decide to make n independent measurements

X1, X2, . . . , Xn of the length. Assume that each Xi has

mean (unbiased measurements) and variance s2.

a. Show that is not an unbiased estimator for m2. [Hint: For

any rv Y, E(Y2)  V(Y)  [E(Y)]2. Apply this with Y  .]

b. For what value of k is the estimator  kS2 unbiased

for 2? [Hint: Compute E(  kS2).]