Probability density of a product of two random variables (discrete and continuous)

Question

If $X$ is a gaussian random variable with $\mu=0$ and $\sigma^2=1$, and $Y$ is a discrete random variable taking the values $\{ {1,-1} \}$ with probability $\frac{1}{2}$ each. Let $Z=XY$, how do I find the CDF and PDF of Z?

Edit: $X$ and $Y$ are independent.

I started by evaluting:

$F_Z(z)=P(Z\leqslant z)=P(XY\leqslant z)=P(X\leqslant\frac{z}{Y})=$ $=P(X\leqslant -z|Y=-1).P(Y=-1) + P(X\leqslant z|Y=1).P(Y=1)$

But I'm not sure about this last expression, it seems incorrect and I can't see where is the problem.

Answer 1

\begin{align} F_Z(z)&=P(Z\leqslant z)\\&=P(XY\leqslant z) \\&=P(XY \le z|Y=-1)P(Y=-1)+ P(XY \le z|Y=1)P(Y=1)\\ &=P(X \ge -z|Y=-1)P(Y=-1)+P(X \le z|Y=1)P(Y=1) \end{align}

If you assume independence, we can further simplify things.

Be careful when you work with negative number and inequalities.