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.
The marginal pdf of U is the classic triangular density...
$\displaystyle f_U(u)=u$ on 0<u<1
$\displaystyle f_U(u)=2-u$ 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.
Let's see your work..............
$\displaystyle f_{XY}(x,y)=1$ on 0<x<1, 0<y<1.
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 $\displaystyle dxdy=rdrd\theta$. The r is the jacobian, it's the determinant of the 2 by 2 matrix of partial derivatives.
I would use calculus, but I also figured you could use the joint CDF.
I've never done that before...
$\displaystyle P(U \le z, V\le b) = P(X+Y \le a, X-Y \le b)=$
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.