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Problem 6. Let X1, . . . ,X denote independent random variables, with X;- N N ([31 + qg, 0'2), i = 1, . . . ,n, Where a is a positive number and 51, 52, 61,0\" are numbers satisfying 1 1 Egg-=0 and T=EZCE>Q Set c-X- ZXj, 32:2112 and a: j=1 (1) Find the distribution of the random vector X. (2) Find the distribution of 31- (3) Find the distribution of 32. (4) Find the distribution of (n 2)c'r2/02. (5) Show that (3'1, 5'2 and 6 are independent. (6) Find the distribution of T1 = Jar1(5'1 51) /& (7) Find the distribution of T2 = W(5152)/0 (8) Find the distribution of n[(11)2 + 'r (2 2) )2]/6'2. (9) Find the distribution of \"7(51 32V (1 + T)5'2 when 51 = '62