38. Each of n specimens is to be weighed twice on the same scale. Let Xi and Yi denote the two observed weights for the ith specimen. Suppose Xi and Yi are independent of one another, each normally distributed with mean value i (the true weight of specimen i) and variance 2

.

a. Show that the maximum likelihood estimator of 2 is

ˆ 2

(Xi  Yi

)2/(4n). [Hint: If z

(z1 z2)/2, then

(zi  z)2
(z1  z2)2

/2.]

b. Is the mle ˆ 2 an unbiased estimator of 2

? Find an unbiased estimator of 2

. [Hint: For any rv Z, E(Z2

)

V(Z) [E(Z)]2

. Apply this to Z
Xi  Yi

.]