2.27 Consider the problem of predicting Y using another variable, X, so that the prediction of Y is some function of X, say g1X2. Suppose that the quality of the prediction is measured by the squared prediction error made on average over all predictions, that is, by E53Y - g1X2426. This exercise provides a non-calculus proof that of all possible prediction functions g, the best prediction is made by the conditional expectation, E1YX2.

a. Let Y n

= E1YX2, and let u = Y - Y n denote its prediction error. Show that E1u2 = 0. (Hint: Use the law of iterated expectations.)

b. Show that E1uX2 = 0.

c. Let Y 

= g1X2 denote a different prediction of Y using X, and let v = Y - Y  denote its error. Show that E31Y - Y  2 24 7 E31Y - Y n 2 24.

[Hint: Let h1X2 = g1X2 - E1YX2, so that v = 3Y - E1YX24 - h1X2.

Derive E1v22.]