Question

Let (X,Y) be a pair of random variables for which the regression function \nu (x) = \mathbb E[Y | X = x] takes the form

\nu (x) = a + bx

for some pair of real numbers (a,b).

What is a random variable \hat{Y} that is a function of X that minimizes

\mathbb E\left[ (Y - \hat{Y})^2 | X = x \right]

over all possible choices of \hat{Y} and for all x? Enter your answer in terms of a, b and the random variable X (capital letter “X").

(Remark: for a clean, quick solution, it may be helpful to review the law of iterated expectations: \mathbb E_{X,Y}[\cdot ] = \mathbb E_{X}[\mathbb E_{Y}[\cdot \; |\; X]], where \mathbb E_{Y}[\cdot |X] denotes the conditional expectation, which is a random variable. Use the insight from the previous exercise.)