$Y=X+\xi X$, $X \sim \operatorname{Unif}[-1, 1]$, $\xi \sim \operatorname{Unif}[0, 1]$. $X$ and $\xi$ are independent.
I need to find $f_{Y\mid X}(y\mid x)$. How to do it? I couldn't find joint probability. Any hints? I think the questions of finding the joint density and the conditional density are equivalent, because it is expressed through each other
The main issue is to derive the joint density
Setting
$Y=X+\xi X$ and $Z=X$ the jacobian is $|J|=1/|z|$ thus the joint density results to me
$$f_{XY}(x, y)=\frac{1}{2|x|}\Big[\mathbb{1}_{[-1;0)}(x)\mathbb{1}_{(2x;x)}(y) + \mathbb{1}_{(0;1]}(x)\mathbb{1}_{(x;2x)}(y) \Big]$$