In practice, it is often desired to predict the value of a variable Y from the known value of some other variableX. For example, a doctor might wish to predict the lifespan Y of someone who smokes X cigarettes a day, or an engineermay require predictions of the tensile strength Y of steel made with concentration X of a certain additive. A linear predictor of Y is anything of the form Ŷ = a + bX the “hat” on Y indicates prediction. A common measure of the quality of a predictor is given by the mean square prediction error, E[(Y – Ŷ)²] .

a. Show that the choices of a and b that minimize mean square prediction error are

where ρ = Corr(X, Y). The resulting expression for Ŷ is often called the best linear predictor of Y, given X.

b. Determine the mean square prediction error for the best linear predictor. How does the value of ρ affect this quantity?

b= a = y b