Please see question below: a and b only

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 (X, - M)2/02 and V = (X - M)?/(o/ n). 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 Z? ~ x2(1)). (b) It is known U and V are independent. Find the distribution of U - V.function of n. Let Y, = X,. Find the limiting distribution of Y,. If 3. Let X1, ... , X, are iid random sample from a Normal distribuition with mean and variance o'. Let S. = LEM,(X, - X)', Complete the following CR -/Offa xax () (a) Let U = EM,(X, - #4)?/o' and V = (X - 14)?/(a/\ )?. Show U follows chi- square distribution with n degrees of freedom and V has limiting distribution V xax ( n ) chi-squre with one degress of freedom. (Hint: if Z ~ N(0, 1), then Z? ~ x?(1)). (5,() (b) It is known U and V are independent. Find the distribution of U - V. (c) Show the limiting distribution of (n - 1)52/62. xty (d) Find E(S2) and Var($2) using the limiting distribution. 4. Let X equal the number out of n = 48 mature aster seeds that will germinate