Problem 1. Show that σ
2
(x) is the best predictor of e
2 given x:
(a) Write down the mean-squared error of a predictor h(x) for e
2
.
(b) What does it mean to be predicting e
2
?
(c) Show that σ
2
(x) minimizes the mean-squared error and is thus the best predictor.
Problem 2. Use y = m(x) + e to show that
Var(y) = Var(m(x)) + σ
2
Problem 3. Show that the conditional variance can be written as
σ
2
(x) = E(y
2
|x) − (E(y|x))2
Problem 4. Let x and y have the joint density f(x, y) = 3
2
(x
2 + y
2
) on 0 ≤ x ≤ 1,
0 ≤ y ≤ 1. Compute the coefficients of the best linear predictor y = α + βx + e.
Compute the conditional mean m(x) = E(y|x). Are the best linear predictor and
conditional mean different?