sum of two dependent random variables

Question

Let $X$ be a cotinuous random variable uniformly distributed over $[-10,10]$. Let $Y$ be a random variable with pdf $f_Y(y) = \frac{1}{40}\ln \frac{20}{|y|}, -20 \leq y \leq 20$.

$X$ and $Y$ ARE NOT independent. How do find the pdf of the sum $X+Y$?

I can't use convolution because of their non-independence.

How do I begin?

Answer 1

Where you can start :

You have that : $Y = f(X)$ and you want the pdf of $X+Y$.

Let $\phi : \mathbb{R}^2 \to \mathbb{R}$ be a smooth bounded function.

Compute $\mathbb{E}[\phi(X+Y,X)]$. That should give you the joint pdf of $(X+Y,X)$. Then you should know how to get the pdf of $X+Y$.