What is the interval of $U$ and $V$?

Question

If the joint distribution of X and Y is given by

$$f_{x,y}(x,y)=2e^{-(x+y)} \hspace{.3cm} I_{(0,y)} (x) I_{(0,\infty)}(y) $$

find the joint distribution of $X$ and $X + Y$.Find the marginal distributions of $X$ and $X + Y$.

Let $U=X$ and $V=X+Y$

doing the Jacobian

$|J|=1 $

so $f(u,v)= 2e^{-v}$, but I don't know what limits $U$ and $V$ have.

Answer 1

Observe that the original support is

$0<x<y<\infty$

that means also

$0<u<v-u<\infty$ or also

$0<2u<v<\infty$

So you join density becomes

$f_{UV}(u,v)=2e^{-v}\mathbb{1}_{(0;\infty)}(u)\mathbb{1}_{(2u;\infty)}(v)=2e^{-v}\mathbb{1}_{(0;\infty)}(v)\mathbb{1}_{(0;\frac{v}{2})}(u)$

Now it is very easy to determine the marginals

$U\sim Exp(2)$

$V\sim Gamma(2;1)$