Problem 1. Let Y1, . . . , Yn be independent random variables each with the distribution N (n, 0'2). Let Y: 1 11 711 TIM: 1 n 7 12, and 32:2(12 Y)? i:1 (a) Show what is the distribution of 17? (b) Show that S2 : [::':l(l \")2 71(17 m2]. (c) From (b) it follows that the 20/1 ,u)2/02 : (n USO/02 + [O7 ,u)2'n,/02]. How does this allow you to deduce that Y and 5'2 are independent? [Hint use Cochran's theorem.] (d) Show what is the distribution of (n 1)S2/0'2? (6) Show what is the distribution of if\" ? El