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.

This is a mixture distribution $$F_X(x)=\frac12F_Z(x)+\frac12F_{-Z}(x)$$ for $x\in \mathbb R$, where the weights $1/2$ correspond to the result of the coin flip (assuming a fair coin, $1/2$ head and $1/2$ tail). Now $$F_{-Z}(x)=P(-Z\le x)=P(Z\ge -x)=1-F_Z(-x)$$ so that you can write $F_X(x)$ for $x\in \mathbb R$ as $$F_X(x)=\frac12F_Z(x)+\frac12\left(1-F_{Z}(-x)\right)$$ If you differentiate the previous equation you get the density of $X$ (so, yes, it has one) $$f_X(x)=\frac12 f_Z(x)+\frac12f_Z(-x)$$ for $x\in \mathbb R$. 
