How to find the joint pdf of $Z=X+Y$ and $W=\frac{X}{Y}$?

Question

I have a joint pdf:

$$ \begin{equation*} f(x,y)=\left\{ \begin{array}{rl} e^{-(x+y)} & \text{if }0< x < y \\ 0 & \text{otherwise} \end{array} \right. \end{equation*}$$

I need to find the joint pdf of $Z$, $W$ where $Z=X+Y$ and $W=\frac{Y}{X}$.

I started by writing $P(Z\le z , W \le w)$, but I am not sure how to continue. Any help?

Answer 1

The joint density of $Z,W$ is, per the method of transformation,

$$f_{Z,W}(z,w)=f_{X,Y}\left(\frac z{1+w},\frac{zw}{1+w}\right)\frac z{(1+w)^2}$$

where $\frac z{(1+w)^2}$ is the (absolute value of the) Jacobian determinant for the transformation from the $(x,y)$ plane to the $(z,w)$ plane.