$X$ is a standard normal random variable and $Y$ is a random variable which takes only the values of either 1 or -1, and $\Bbb E[Y]=1$. $X$ and $Y$ are independent. What is the distribution of $Z=XY$ and what is the covariance of $X$ and $Z$?
Here is what I have: $$Y = \begin{cases} 1, & \text{with probability $p$} \\ -1, & \text{with probability $1-p$} \end{cases}$$ $$Z=XY=\begin{cases} X, & \text{with probability $p$} \\ -X, & \text{with probability $1-p$} \end{cases}$$ So therefore if $\text {Cov}(X,Z)=\Bbb E[XZ]=\Bbb E[X]\Bbb E[Z]$ we need $\Bbb E[XZ]$. Clearly $\Bbb E[X]\Bbb E[Z]$ must be zero because $X$ is standard normal. I also take $\Bbb E[Z]=\Bbb E[X]\Bbb E[Y]=0$ due to independence. So what is $\Bbb E[XZ]$?
I attempted to find the distribution of $XZ$ and got $$Z=XY=\begin{cases} X^2, & \text{with probability $p$} \\ -X^2, & \text{with probability $1-p$} \end{cases}$$ Is this right? If so, how can we get the expected value? If not, where am I going wrong?