2. Normal Sample Mean and Sample Variance, Part 1 Let X1, X2, , Xn be i.i.d. with mean u and variance 0'2. Let denote the sample mean. In Homework 9, you constructed a random variable 52: 1 ion X? :21 n 1 . called the sample variance. Before proceeding, please review Homework 9, Question 1 in its entirety. a) For 1 S i S n let D1 = X; X. Find Cov(D,~, X). b) Now assume in addition that X1, X2, , X,1 are i.i.d. normal (,u, 0-2). What is the joint distribution of X, D1, D2, , Dn_1? Explain why D,l isn't on the list. c) True or false (justify your answer): The sample mean and sample variance of an i.i.d. normal sample are independent of each other. 3. Normal Sample Mean and Sample Variance, Part 2 a) Let R have the chisquared distribution with :1 degrees of freedom. What is the mgf of R? b) For R as in Part (a), suppose R = V + W where V and W are independent and V has the chi-squared distribution with m 0. Let S2 be the "sample variance" defined by .92: 1 ionXV "1 1:1 Find a constant c such that 052 has a chisquared distribution. Provide the degrees of freedom. [Use Parts (b) and (c) as well as the result of the previous exercise]