Coin flipping combined with exponential distribution Let $Z\sim \exp(1)$. Let $X$ be a new random variable (rv) defined as follows: We flip a coin. If we get head, than $X=Z$, and if we get tail than $X=-Z$.
I'm trying to figure whether $X$ is a discrete, continuous or mixed type rv, and to calculate its CDF and PDF (if it has one), but couldn't arrive at a solution.
Any help will be greatly appreciated.

Preassumed that the coin is fair:
$$F_{X}\left(x\right)=P\left(X\leq x\right)=P\left(X\leq x\mid\text{Head}\right)P\left(\text{Head}\right)+P\left(X\leq x\mid\text{Tail}\right)P\left(\text{Tail}\right)=P\left(Z\leq x\right)\frac{1}{2}+P\left(-Z\leq x\right)\frac{1}{2}$$
et cetera.
