a. Express the ith residual Yi - Yi (where Yi = 0 + 1xi) in the form

Question:

a. Express the ith residual Yi - Ŷi (where Ŷi = β̂0 + β̂1xi) in the form ΣjYj, a linear function of the Yj's. Then use rules of variance to verify that V(Yi - Ŷ) id is given by Expression (13.2).
b. It can be shown that Ŷi and Yi - Ŷi (the ith predicted value and residual) are independent of one another. Use this fact, the relation Yi = Ŷi + (Yi - Ŷi) id, and the expression for V(Ŷ) d from Section 12.4 to again verify Expression (13.2).
c. As xi moves farther away from x, what happens to V(Ŷi) id and to V(Y - Ŷi) id?