Consider the scenario in which
U = g(X,Y) = X2
And
V = h(X, Y) = X + Y.
Suppose the joint density fX,Y (x,y) of X and Y is Uniform on the square where 0 ‰¤ X,Y ‰¤ 1. In other words,
fX,Y(x,y) = 1
if 0 ‰¤ x ‰¤ 1 and 0 ‰¤ y ‰¤ 1
and fX,Y(x,y) = 0 otherwise.
a. Find (/(x g (x,y), (/(g(x,y), (/(h(x,y), (/(y h(x,y)
b. Now write the joint density of U and V:

using the values derived above, and using the fact that fX,Y(x,y) = 1 in the region where X, Y is defined. Make sure that, in your expression of fU,V(u,v), you convert all x's and y's to u's and v's in an appropriate way.

fx.y(r, y) Suv (u, v) =