Joint density of $X,Y$ is $$f_{X,Y}(x,y) = a^2e^{-ay},$$
and I know that $X + W = Y$, where $0 < x < y < \infty$
I want to determine the joint density of $X$ and $W$. I know I can let $X = X$, $W = Y - X$, can through calculate the Jacobian and through solving the following equation, I can get the density of $X$ and $W$: $$f_{Y_1,Y_2}(y_1, y_2) = \frac{f_{X_1,X_2}(x_1, x_2)}{|\mathcal J (x_1, x_2)|}$$ where $x_1 = h_1(y_1, y_2), x_2 = h_2(y_1, y_2)$
However, I am confused in terms of determining the region $X$, and $W$. I know $0<x<y$, which means that $0< x < x+w$. I can get $w > 0$, but I am confused in terms of determining the upper region of $X$. Is it just $x >0$? But $X$ also needs to be smaller than $X + W$?