Given that $X\sim N(0,1)$ and $Z\sim \operatorname{Unif}(\{\pm1\})$. Prove that $Y = XZ$ is a standard Gaussian distribution.
My approach:
I started trying to find the pdf of them. I found that:
$$f_Y(y)=\int_{-\infty}^\infty f_Z(z)f_X\left(\frac{y}{x}\right) \frac{1}{x} \, dx=\int_{-\infty}^\infty \frac{\exp(-\frac{(x/y)^2}{2})}{x\sqrt{2\pi}}\,dx$$
and I can't seem to find finish the calculation.
So I moved on to the MGF approach with:
$$M_{ZX}(t)=E[e^{zxt}] = \int_{-1}^{1}\int_{-\infty}^\infty \exp(zxt)\frac{\exp(-x^2/2)}{\sqrt{2\pi}}\frac{1}{2} \, dx \, dz = \frac{1}{2}\int_{-1}^{1}\exp(\frac{1}{2}(zt)^2) \, dz$$
and again, I can't finish it…
All of the answers seems to lead up to erfi functions…
edit: additional info: $Z$ and $X$ are independent of each other
Best Answer
Reading the text I understand that Z is a discrete rv taking the values $\pm1$ with probability $\frac{1}{2}$ so,
simply observe that $Y=-X\sim N(0;1)$ (it's trivial but you can derive the result with fundamental transformation theorem)
so $f_Z(z)=1/2\phi(z)+1/2\phi(z)=\phi(z)$