Problem 1 [15 pt] Independence of B and 62 Consider a linear regression model Y =XB+E where e = (61, . .., En) and ci N(0, 62) Recall that 1 1 2-p-1 Y ( I - H)Y = n- p- 1 Y ( I - H)2Y and B = (X X) -1 xTy 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 (I - H) Y are independent. 3. Recall that in the class we considered the ratio 1 T = aTB - aT B* aTB - aT BX ova ( X X )-la ova (X X)-la o/0 here a = (do, al, . . ., ap) is a fixed vector. Argue from parts 1 and 2 that aTB-aT B* ovaT(XTX)-la and o /o are independent