Conditional density and Uniform distribution

Question

$Y=X+\xi X$, $X \sim \operatorname{Unif}[-1, 1]$, $\xi \sim \operatorname{Unif}[0, 1]$. $X$ 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

Answer 1

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

$$f_{XY}(x, y)=\frac{1}{2|x|}\Big[\mathbb{1}_{[-1;0)}(x)\mathbb{1}_{(2x;x)}(y) + \mathbb{1}_{(0;1]}(x)\mathbb{1}_{(x;2x)}(y) \Big]$$