Question 2 (13 Marks): use of a software is not recom- mended Let X = [X1,X2]T ~ N2(,u,2) be a bivariate normal vector, where a = 1 c c 2 . Let E be positive denite and [#1, #le and 2 = d2(X,.u) = (X - #)TE'1(X - H)- (a) (3 Marks) Obtain the distribution of 2'1/2(X ,u). Show your working. (b) (3 Marks) Let ([X11,X12]T,[X21,X22]T,...,[Xn1,Xn2]T) be H. inde pendent copies of X and f = [4321, X2]T be the vector of averages 21 = 2X1] and X2 = 2X12. i=1 i=1 Let F(:c) be the cumulative distribution function (CDF) of nd2(}-{, p). Identify F (2:) including its parameter(s). Give your reasoning. (c) (3 Marks) Let the signicance level a = 0.1 and F(4.61) = 10:, where F(m) is dened in (b). Suppose that E is known. The sample of size n = 4 is randomly sampled from the distribution of X and the sample mean vector i = [0.3,O]T. Let H0 : a = [0, 0]T and H1 : [J 76 [O,O]T. Determine the interval for all c for which the hypothesis H9 will not be rejected at the signicance level or = 0.1 (d) (2 Marks) Now suppose that S is unknown. What is the distribution of 710120? , p) ? (e) (2 Marks) In this framework, what is the decision rule for rejecting the null hypothesis H0 : u = [0, OF