I have encountered some problems when solving problems of bivariate continuous random variables finding the new support. For example,
f(x,y) =\begin{cases} 2e^{-x-2y}, & \ {0<x<y<\infty} \\ 0, & \text{else where} \end{cases}
To find the distributions of U and V when where U = Y - X, V = Y, I found the joint pdf of U and V as the following:
f(u,v)= \begin{cases} 2e^{-3v+u}, & \in \color{red}{ T=\{(u,v): 0<u<v,v < u+v, v<\infty \}} \\ 0, & \text{else where} \end{cases}
My challenge is to find the correct T/range as support for the joint density function of U and V and, then I will use it to find the marginals of each. What is also challenging is finding the support (range) of these marginals as well.
I have looked up for similar questions but it does not seem that I have grasped the method to do so. Is there a more intuitive way to think about it and solve such problems?
Also, I would appreciate more references to read and practice on this particular part of the problem.
Take the original constraint $0<x<y<\infty$ and write it in terms of $u$ and $v$ to get $0 < v-u < v < \infty$. Rewriting these last inequalities gives $0 < u < v < \infty$. This is what you have (note that your $v < u+v$ inequality is redundant with $0 < u$).
To find the support of the marginals, just take the constraints from the joint PDF and ignore the other variable. So $0 < u < \infty$ and $0 < v < \infty$.