I am stuck on a question with two parts :
For $f(x,y) = \begin{cases} a^2 e^{-ay} & 0 \le x \le y, \\ 0 & \text{otherwise} \end{cases}$
a) Compute the distribution function and density of $Z = X+Y$
b) Find the joint distribution function and the densty of $(Y, X+Y)$
I have been able to calculate part a) by integrating the function by taking limits for $x$ from $0$ to $z/2$ and for $y$ $x$ to $z-x$ and I get the correct solution.
For part b) I am applying the theorem that $f(y_1,y_2) = f(x_1(y_1,y_2),x_2(y_1,y_2))|J(y_1,y_2)|$ where $J$ is the Jacobian.
But I am unsure about the limits for the integral. How do I proceed?