We have joint probability density function (PDF) of X and Y
$f(x,y) = \begin{cases} e^{-x} & \text{for x > 0 ,y $\in$ (0,1), } \\ 0 & \text{in other case.} \end{cases}$
Let $Z = X + 2Y$. Find the PDF of (Z,X) in $\{(z,x)\in R^{2}:0<x<z<2+x\}$
So we don't assume that X and Y are independent. I found marginal PDFs of X and Y from their joint PDF.
$f_{X}(x)=\begin{cases} e^{-x} & \text{for x > 0} \\ 0 & \text{in other case} \end{cases}$
$f_{Y}(y)=\begin{cases} 1 & \text{for y $\in$ (0,1)} \\ 0 & \text{in other case} \end{cases}$
and $f_{Y}(y)*f_{X}(x)=f(x,y)$ so X and Y are indepedent so maybe I can find Z somehow from convolution theorem. And I don't know what to do next. Especially when i have to find pdf of (Z,X).