Consider a linear regression model Y = XB +E where e = (61, . . ., En) and e; "" N(0, 02) Recall that 1 n - p-1 YT (I - H)Y = n-p- 1 YT (I - H)2Y and B = ( X X) -1xTy In order to show that S and 62 are independent, it suffices to show that B and (I - H) Y are independent. We will argue that in this problem. 1. Argue that Y is also a normal random variable, and write down it's mean vector and covariance matrix. 2. Suppose Y ~ N(u, E), then the vectors AY and BY are independent when AEB = 0. Here A and B are two fixed matrices. Using this result show that B and (1 - H) Y are independent. 3. Recall that in the class we considered the ratio a TB - aT B * a TB - aT B* T = ova ( X X )-la ova(X X )-la o/0 here a = (ao, a1, .. ., ap) is a fixed vector. Argue from parts 1 and 2 that aTB-aTB* ovaT(XT X) -la and o/o are independent