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

Using a long rod that has length , you are going to lay outa square plot in which the length of each side is . Thus thearea of the plot will be 2. However, you do not know thevalue of , so you decide to make n independent measurementsX1, X2, . . . , Xn of the length. Assume that each Xi hasmean (unbiased measurements) and variance s2.a. Show that is not an unbiased estimator for m2. [Hint: Forany rv Y, E(Y2) V(Y) [E(Y)]2. Apply this with Y .]b. For what value of k is the estimator kS2 unbiasedfor 2? [Hint: Compute E( kS2).]

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