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

).]