7. Let (X, Y ) be a random pair and let Yb = baX +bb be the linear regression of Y based on X. For notation, we use r = r(X, Y ) to denote the correlation between X and Y and use D = Y Yb to denote the residual. We write E(X) = X, Var(X) = X, E(Y ) = Y , and Var(Y ) = Y .

(a) [6 points] Show that D = Y Yb and Yb are uncorrelated, i.e. r(D, Yb) = 0.

(b) [6 points] Show that Var(Y ) can be decomposed into Var(Y ) = Var(Yb) + Var(D).