Let f (x, y) = 1/8, 0 ≤ y ≤ 4, y ≤ x ≤ y + 2, be the joint pdf of X and Y.

(a) Sketch the region for which f (x, y) > 0.

(b) Find f_X(x), the marginal pdf of X.

(c) Find f_Y(y), the marginal pdf of Y.

(d) Determine h(y | x), the conditional pdf of Y, given that X = x.

(e) Determine g(x | y), the conditional pdf of X, given that Y = y.

(f) Compute E(Y | x), the conditional mean of Y, given that X = x.

(g) Compute E(X | y), the conditional mean of X, given that Y = y.

(h) Graph y = E(Y | x) on your sketch in part (a). Is y = E(Y | x) linear?

(i) Graph x = E(X | y) on your sketch in part (a). Is x = E(X | y) linear?