1. Let yˆ = E (y|x) be the conditional expectation of y given x. Prove that E{(y - yˆ)² } ≤ E{(y - π)² }, where π = π(x) is any other function of x. Show that E {x(y - yˆ)} = 0 and give an interpretation of this condition.
Demonstrate that, if E (y|x) is a linear function of x, then we have E(y|x) = E(y) + β { x - E(x)} , where β = C (x, y)/ V (x).