Let X and Y have a continuous joint pdf

f(x, y) = e^y , 0 < x < y.

a) Show that the marginal pdf of X is fX (x) = e^x, x > 0 and use the marginal pdf to calculate E[X].

b) Show that the marginal pdf of Y is fY (y) = ye^y, y > 0 and use the marginal pdf to calculate E[Y ].

c) Find the conditional distribution of X given . What is the name of the conditional distribu-tion?

d) Obtain the conditional expectation E[X|Y = y].

e) Use the results in (b) and (d) to calculate E[X]. Compare the result to that you obtain in (a).

f) Show that E[XY ] = 3 and Cov(X, Y ) > 0.

g) Are X and Y correlated? Explain your answer.

h) Are X and Y independent? Explain.