3. Let X1, . .. , Xn are iid random sample from a Normal distribuition with mean / amd variance o'. Let S = , Et(Xi - X). Complete the following: (a) Let U = En(Xi - M)2/02 and V = (X - M)2/(o/Vn). Show U follows chi- square distribution with n degrees of freedom and V has limiting distribution chi-squre with one degress of freedom. (Hint: if Z ~ N(0, 1), then Z2 ~ x2(1)). (b) It is known U and V are independent. Find the distribution of U - V. (c) Show the limiting distribution of (n - 1) $2/62. (d) Find E(S?) and Var(S?)