What is the distribution of the product of a Bernoulli & a normal random variable? I am struggling to find a solution on finding the distribution of following random variable:
$$Y = Z \cdot  |X|$$
here, $Z$ is a random variable takes 1 or -1 with equal probability, and $X$ is a standard normal variate, and $|\cdot|$ denotes absolute value.
Can somebody help me with some pointer from where I should start?
Thanks for your help.
 A: $-X$ has the same distribution as $X$ since its density is symmetric about the origin, and $Z$ is likewise symmetric, therefore the result is ... yet another normal random variable.
It's instructive to ponder how $Y$ is impacted by changes in the parameter $p=\mathrm P(Z=1)$ of the Bernoulli random variable $Z$. Here is a plot of $Y$ as $p$ runs from $0$ to $1$:

Can you mentally confirm this animation by imagining $Y$ for $p=0$, $p=0.5$, and $p=1$, then doing a little interpolation?
A: May I suggest you start from first principles?  You seek the distribution of $Y$, so you should be asking yourself about the chance that $Y \le t$ for some arbitrary real value $t$.  To handle the discreteness of $Z$, consider enumerating its possible values:
$$\Pr[Y \le t] = \Pr[Z\ |X| \le t] = \Pr[|X| \le t \text{ and }Z=1] + \Pr[-|X| \le t \text{ and } Z=-1].$$
Because you are assuming $X$ and $Z$ independent, the joint probabilities (connected by "$\text{and}$") are obtained by multiplication.  The rest now is straightforward.
By doing the computations graphically (use a sketch of the PDF of $X$) you will likely note some opportunities for simplification of the answer; it reduces to a very simple expression in terms of the cumulative distribution function of $X$ itself.
A: Here, I have done following calculation:
P(Y <= y)
=P(Z*mod(X) <= y)
=0.5P(mod(X) <= y) + 0.5P(-mod(X) <= y)
=0.5*[ P(-y <= X <= y) + P(mod(X) >= -y) ]
=0.5*[ 2Phi(y) - 1 + 1 - P(mod(X) < -y) ]
=0.5*[ 2Phi(y) - P(y <= X <= -y) ]
=Phi(y)       because, 2nd component is probability of impossible event
Therefore Y have standard normal distribution. 
Is my calculation is correct?
