Q:

Is this a normal distribution?

Given $Z\sim N(0,1)$, a coin is flipped.
Another random variable
$$
W = \begin{cases} \phantom{-}Z & \text{if heads}, \\
-Z & \text{if tails}.
\end{cases}
$$
Is $W$ normal? 
I just am not really sure how to start this problem, any help would be greatly appreciated.

A:

It is normal.
\begin{align}
\Pr(W\le w) & = \Pr(W\le w\mid\text{heads})\Pr(\text{heads}) + \Pr(W\le w\mid\text{tails})\Pr(\text{tails}) \\[8pt]
& = \Pr(W\le w\mid\text{heads})\Pr(\text{heads}) + \Pr(Z\ge -w \mid \text{tails}) \Pr(\text{tails}) \\[8pt]
& = \Phi(w)\cdot\frac12 + \left(1-\Phi(-w)\right)\cdot\frac12 \\[8pt]
& = \Phi(w)\cdot\frac12 + \Phi(w)\cdot\frac12 \\[8pt]
& = \Phi(w).
\end{align}

