Suppose $X$ and $Y$ are continuous random variables with joint p.d.f.
$$f(x,y) = e^{-y},\,\, 0<x<y <\infty$$
(a) Find the joint p.d.f. of $U=X+Y$ and $V=X$. Be sure to specify the support of $(U,V)$.
(b) Find the marginal p.d.f. of $U$ and the marginal p.d.f. of $V$. Be sure to specify their support.
I can't figure out what I am doing wrong with this question. So far, I have gotten that the support for $U$ and $V$ is $0<v<u<\infty$, the Jacobean matrix has determinant $-1$ and that the joint p.d.f for part a) is $e^{v-u}$ but this p.d.f doesn't make sense when I try to find the marginals. Could someone help guide me in the right direction?
First I checked if it is indeed a density $$\int_0^\infty\int_0^ye^{-y}dxdy=1$$
and $X,Y$ are not independent.So I proceeded as you already did
$(1)U=g_1(X,Y)=X+Y$ and $(2)V=g_2(X,Y)=X$ then $J=\begin{bmatrix}\frac{\partial u}{\partial x}&&\frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x}&&\frac{\partial v}{\partial y}\end{bmatrix}=-1$ hence $|J|^{-1}=1$
You know that $Y=U-V$ from $(1)$ and $(2)$ $$f_{U,V}(u,v)=f_{X,Y}(g_1^{-1}(u,v),g_2^{-1}(u,v))|J|=e^{v-u}$$
You know that $$0<x<y\iff0<v<u-v\iff0<2v<u$$ hence $$f_U(u)=\int_0^{u/2} e^{v-u}dv=(e^{-u/2}-e^{-u})\,I_{[0,\infty)}(u)$$ $$f_V(v)=\int_{2v}^\infty e^{v-u}du=e^{-v}I_{[0,\infty)}(v)$$