# Transforming sum of random variables and conditional expectation

Homework warning

I've been having a lot of trouble with questions where we transform a random variable with a sum of other random variables.

I'll give an example and what I've tried to do (this is not an actual question from my homework but is tangentially related).

Say $$X=\begin{cases}-1 , \text{with probability of } \frac{1}{2} \\ 1 , \text{with probability of } \frac{1}{2}\end{cases}$$ and $$Y = X + Z, Z \sim Uniform[-1,1],\text{ and } X\perp \!\!\! \perp Z$$ If I'm given $$Y=y$$ I'm completely lost on how I would find $$p(X \mid y) \text{ and similarly } E(X|y)$$

The method I used in the past would be $$F_{X \mid Y}(x \mid y) = P(X \leq x \mid y) = P(y - z \leq x \mid y) = P(z \leq y - x \mid y)=F_{Z \mid Y}(y-x \mid y) \\ \Rightarrow p_{X\mid Y}(x \mid y) = \frac{d}{dz}(F_{Z \mid Y}(y-x \mid y))$$ But at that point I get stuck as I don't know if it's correct to assume that $$F_{Z \mid Y}(y-x \mid y)) = F_Z(y-x)$$ or if there's another assumption I should make to transform this into the pdf I want or if that's even the correct method for finding the pdf I want. I know this question seems like it's all over the place and I apologize in advance for that as I'm very lost at the moment in what route I should be taking.

If there's anyway I can clarify things please comment and I'll try my best