10. 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 2

.

a. Show that X2 is not an unbiased estimator for 2

.

[Hint: For any rv Y, E(Y2

)
V(Y) [E(Y)]2

. Apply this with Y
X.]

b. For what value of k is the estimator X2  kS2 unbiased for 2

? [Hint: Compute E(X2  kS2

).]