Jacobian Transformation p.d.f > 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