Q1. (Example 7.9 7.10) Suppose X1, . . ., Xn ~ N(u, 2). Consider the sample variance estimator of 02, S'2 n - (Xi - X)2 (a) Show that S2 an unbiased estimator of o2? (b) Show that the MSE($2) is MSE(S2) = Var(S?) = 204 n - 1 (c) Consider the MLE/MME of o2, 82 = = L(Xi - X)? Show that the MSE of 82 is MSE(2) = 2(n - 1)04 2 = (2n - 1)04 n2 + n2 Q2. (Cramer-Rao Inequality). Let X1, . .., Xn ~ f(.10). Let 0 = W(X1, . .., Xn) = W(X) be an unbiased estimator of 0. Assume that f(x|0) satisfies the following two conditions i . " Eow ( X ) - / 20 [ W ( x ) ; ( x 1 0 ) ] dx ii. For each 0, the variance of 0 = W(X1, . . ., Xn) is finite, VaroW (X)