Let X be the height of a randomly chosen adult man, and Y be his fathers height, where X and Y have been standardized to have mean 0 and standard deviation 1. Suppose that (X, Y ) is Bivariate Normal, with X, Y N (0, 1) and Corr(X, Y ) = .2

(a) Let y = ax + b be the equation of the best line for predicting Y from X (in the sense of minimizing the mean squared error), e.g., if we were to observe X = 1.3 then we would predict that Y is 1.3a + b. Now suppose that we want to use Y to predict X, rather than using X to predict Y . Give and explain an intuitive guess for what the slope is of the best line for predicting X from Y .

(b) Find a constant c (in terms of ) and an r.v. V such that Y = cX + V , with V independent of X. Hint: Start by finding c such that Cov(X, Y cX) = 0.

(c) Find a constant d (in terms of ) and an r.v. W such that X = dY + W, with W independent of Y .

(d) Find E(Y |X) and E(X|Y ).

(e) Reconcile (a) and (d), giving a clear and correct intuitive explanation.