In practice, it is often desired to predict the value of a variable y from the known value of some other variable, x. for example, a doctor might wish to predict the lifespan y of someone who smokes x cigarettes a day, or an engineer may 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 ^y ¼ a þ 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' ^y $ %2 h i 4.2 expected values, covariance, and correlation 263 (a) show that the choices of a and b that minimize mean square prediction error are b ¼ rho ( σy σx a ¼ μy ' b ( μx where rho ¼ corr(x, y). the resulting expression for^y is often called the best linear predictor of y, given x. [hint: expand the expression for mean square prediction error, apply linearity of expectation, and then use calculus.] (b) determine the mean square prediction error for the best linear predictor. how does the value of rho affect this quantity?