Let $X \sim \mathcal{N}(0, 1)$ and $Y$ be a random variable independent of $X$ such that \begin{align*} P(Y=y) = \begin{cases} \frac{1}{2} & y = -1\\ \frac{1}{2} & y = 1\\ 0 & otherwise \end{cases} \end{align*} If $Z = XY$, are $Z$ and $X$ independent?
I've found that the correlation between $X$ and $Z$ is 0; however, I know that this does not say anything about their independence.
From the problem, it is quite obvious that the two variables are not independent; however, I am having trouble showing this formally. I want to find some way to show $f_{X,Z}(x,z)\neq f_X(x) \cdot f_Z(z)$ for some $(x, z)$, but I'm not sure what the joint density function actually is.
Since both $X$ and $Z$ follow $\mathcal{N}(0,1)$, I assume that the density function for both variables is the normal density. What, then, would be the joint density?