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

a. Show that the maximum likelihood estimator of o is =(X, Y)/(4n). [Hint: If (2, + z)/2, then (-3)=(-2)/2.]

b. Is the mle o an unbiased estimator of o? Find an unbiased estimator of o2. [Hint: For any rv Z, E(Z) = V(Z) + [E(Z)]. Apply this to Z = X - Y]