Let {X1, X2, ... , Xn} be a random sample from the normal distribution n(µ, o²). A com used unbiased estimator for o? is the sample variance defined by n 1 S? : E(X; – X)². n - i=1 (a) Find the MSE (Mean squared error) of S². (b) Another commonly adopted definition of the sample variance is given by E(X; – X)² п — S². n –1 s2 SD - Find the MSE of SD² as an estimator of o? and compare it to S². Which one is better? (c) Suppose that we seek for an estimator that takes the form ô? constant. Find the value ofc such that ô² has the smallest MSE among all the estimators form of cS?. cS?, where c is some p-

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