1. ## Joint PDF

Any help appreciated! Thanks in advance!

Suppose that X and Y are independent uniform U(0, 1) random variables.

(i) Find the joint PDF of U and V , fU,V (u, v), defined by U = X + Y and
V = X Y .

(ii) Obtain the marginal PDF of U.

2. The marginal pdf of U is the classic triangular density...

$f_U(u)=u$ on 0<u<1

$f_U(u)=2-u$ on 1<u<2

and zero elsewhere.
This can be done directly via CDFs.

As for the joint density of U and V, you need to use calc 3 and jacobians.

$f_{XY}(x,y)=1$ on 0<x<1, 0<y<1.

3. I'm not sure what you mean by "calc 3" and Jacobians

This question isn't remotely similar to any examples we were given when we were learning the stuff with Jacobians

4. You can get the marginals with the CDF technique but I think you have to do a two to two change of variables like you do in calculus with polar coordinates to get the joint density of U and V.
Do recall in polar, that $dxdy=rdrd\theta$. The r is the jacobian, it's the determinant of the 2 by 2 matrix of partial derivatives.

5. Still don't know what calc 3 is... an American maths class, maybe?

This is seeming a little more complicated than necessary... darn lecturer's not here until Friday either. Oh well.

6. I would use calculus, but I also figured you could use the joint CDF.
I've never done that before...

$P(U \le z, V\le b) = P(X+Y \le a, X-Y \le b)=$

the density of X,Y is one on the unit square.
You will need to draw and consider the various cases of a and b.