Let X1, X2, , X\" be i.i.d. with mean [,4 and variance 02. Let denote the sample mean. In Homework 9, you constructed a random variable 1 \" - 52 = 1 20g. 2()2 n 1:1 called the sample variance. Before proceeding, please review Homework 9, Question 1 in its entirety. a)For15i$ in let D,- = X,- X. Find Cav(Di, X). b) Now assume in addition that X1, X2, , X,1 are i.i.d. normal (.11, 0'2). What is the joint distribution of X, D1, D2, , Dn_1? Explain why D,1 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. a) Let R have the chisquared distribution with 11 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 chisquared distribution with m 0. Let S2 be the "sample variance" defined by 1 n 52 = X- X 2 n _ 1 g; . ) Find a constant c such that 032 has a chisquared distribution. Provide the degrees of freedom