Let X and Y be two continuous random variables with joint probability density function (PDF):
f_{X,Y}(x,y) = { 6xy 0 <= x <= 1, 0 <= y <= sqrt(x), 0 otherwise.
(a) Sketch the support of f_{X,Y}(x, y) in the (X, Y) plane. In particular, shade the area in which f_{X,Y}(x, y) > 0.
(b) Find f_X(x) and f_Y(y).
(c) Are X and Y independent?
(d) Find the conditional PDF f_{X|Y}(x|y).
(e) Find E[X|Y = y], for 0 <= y <= 1.
(f) Find Var[X|Y = y], for 0 <= y <= 1.