I'm working on the following problem:
Consider random variables $X$ and $Y$ such that exactly one of them is equal to $0$. The other then takes the value $1$ or $-1$ with equal probability (ex: if $X = 0$, then $Y = 1$ or $Y = -1$ with equal probability).
Are $X$ and $Y$ correlated?
Here is my work so far:
By definition,
$$ \text{corr}(X, Y) = \frac{\text{cov}(X, Y)}{\sigma_x \sigma_y}. $$
By definition, $\text{cov}(X, Y) = \mathbb{E}\left[ (X - \mathbb{E}[X])(Y - \mathbb{E}[Y]) \right]$. Let us compute $\mathbb{E}[X]$. We can write
$$ \mathbb{E}[X] = \mathbb{E}[X \mid X = 0] \mathbb{P}(X = 0) + \mathbb{E}[X \mid X \neq 0] \mathbb{P}(X \neq 0). $$
From the problem statement, we know that $\mathbb{P}(X = 0) = \mathbb{P}(X \neq 0) = 1/2$ and $\mathbb{E}[X \mid X \neq 0] = 0$. It is also clear that $\mathbb{E}[X \mid X = 0] = 0$. Thus, the covariance expression reduces to $\text{cov}(X, Y) = \mathbb{E}[XY]$.
My question: how do I proceed from here? I just want to know whether or not the two random variables are correlated, which reduces to determining whether or not $\text{cov}(X, Y) = 0$. However, I'm not sure how to go about evaluating what I have left.
Any ideas?