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).]
