Proving $Y = WX$ and $X$ are uncorrelated

Question

From my previous question here, I am able to prove $Y \sim \mathcal{N}(0,1)$, where PMF of $W$ is

$$ P(W = w) = \begin{cases} \frac{1}{2} \hspace{3mm} \text{if} \hspace{3mm} w = \pm1 \\ 0 \hspace{3mm} \text{otherwise}. \end{cases} $$ and $X\sim\mathcal{N}(0, 1)$, independent of $W$. I need to show that $X$ and $Y$ are uncorrelated. I have tried evaluating $\mathbb{E}XY = \mathbb EWX^2$, which needed the distribution of $X^2$, which I have evaluated to be $\frac{1}{2\sqrt x}\frac{1}{\sqrt{2\pi}}e^{-\frac{x}{2}}, \hspace{3mm} x \geq 0$. How do I use this to prove uncorrelatedness?

Answer 1

Because $X$ is independent of $W$, $$E[WX^2] = E[W] E[X^2] = 0.$$