Let the pair (X, Y ) have a uniform density over the interior

of the triangle with vertices at (0, 0), (2, 0) and (1, 2), that is, the density is the positive

constant in the interior, and zero otherwise.

a) Find the conditional density of Y given X.

b) Find the conditional expectation E[Y |X = 1].

c) Find the conditional variance V ar[Y |X = 1].

d) Assume now that (X, Y ) has the above uniform density with probability

1/4, and otherwise has the analogous uniform density over the interior of the triangle with

vertices at (0, 0), (2, 0) and (1, 4). What is E[Y |X = 1] equal to now?