The distribution of the product of Gaussian variable and Rademacher variable.

Question

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same probability.

My question is:

For a new random variable of $Z=XY$, What is the probability density function? Or can we have some concentration inequalities for $Z$ (e.g., show how $Z$ approaches its expectation, etc.)?

Answer 1

If $X$ and $Y$ are independent and $Y$ takes values in $\{-1,1\}$ while the distribution of $X$ is symmetric about $0$, then $XY$ has the same distribution as $X$. Hint: condition on $Y$.

Answer 2

Since $X$ has a symmetic distribution about $0$, you have $P(X-0 \le x) = P(0-X \le x)$, i.e. $P(X \le x) = P(-X \le x)$.

Now consider the cumulative distribution function $P(Z \le x)$.

This is $P(XY \le x)=P(XY \le x|Y=1)P(Y=1)+P(XY \le x|Y=-1)P(Y=-1)$ which is $P(X \le x|Y=1)P(Y=1)+P(-X \le x|Y=-1)P(Y=-1)$ which by the symmetry and independence is $P(X\le x)(P(Y=1)+P(Y=-1))=P(X\le x)$ the cumulative distribution function of $X$.

So $Z$ has the same distribution as $X$.

This uses the symmetry of the distribution of $X$ about $0$, $X$ and $Y$ being independent, and $Y$ taking the values $-1$ and $1$. It does not use the normal distribution, or $Y$ taking its values with equal probability.