12. Using a long rod that has length \(\mu\), you are going to lay out a square plot in which the length of
each side is \(\mu\). Thus the area of the plot will be \(\mu^2\). However, you do not know the value of \(\mu\), so
you decide to make n independent measurements \(X_1, X_2, \dots, X_n\) of the length. Assume that each \(X_i\)
has mean \(\mu\) (unbiased measurements) and variance \(\sigma^2\).
(a) Show that \(\bar{X}^2\) is not an unbiased estimator for \(\mu^2\). [Hint: Apply the hint from Exercises 8 and
10 with \(Y = \bar{X}\).
(b) For what value of k is the estimator \(\bar{X}^2 - kS^2\) unbiased for \(\mu^2\)? [Hint: Compute \(E(\bar{X}^2 - kS^2)\),
using the result of Exercise 10(d).]
