Double exponential distribution Let $\zeta$ and $\eta$ be independent random variable with $\exp(\lambda)$ distribution. What is  the distribution of $Z=|\zeta-\eta|$ . I am trying to calculate it by finding $\Pr(\zeta-\eta>x)$, and $\Pr(\eta-\zeta>x)$. Thank you in advance
 A: Since $|\zeta-\eta|\ge0$ almost surely, the probabilities $\Pr(|\zeta-\eta|> x)$ for every nonnegative $x$ fully determine the distribution of $|\zeta-\eta|$. Now, let $x$ be nonnegative. The event $[|\zeta-\eta|>x]$ is the disjoint union of the events $[\zeta-\eta> x]$ and $[\eta-\zeta> x]$. By symmetry, these two events have the same probability, so let us compute the probability of the first one. 
For every $y$, $\Pr(\zeta>y)=\mathrm e^{-\lambda y}$, hence, by independence of $\eta$ and $\zeta$,
$$
\Pr(\zeta-\eta>x\mid\eta)=\mathrm e^{-\lambda (x+\eta)}.
$$
Integrating this with respect to the distribution of $\eta$ yields
$$
\Pr(\zeta-\eta>x)=\mathrm e^{-\lambda x}\,\mathrm E(\mathrm e^{-\lambda \eta})=\mathrm e^{-\lambda x}\int\limits_0^{+\infty}\mathrm e^{-\lambda t}\lambda \mathrm e^{-\lambda t}\text{d}t=\frac12\mathrm e^{-\lambda x}.
$$
Summing up the contributions of $[\zeta-\eta> x]$ and $[\eta-\zeta> x]$, one gets $\Pr(|\zeta-\eta|>x)=\mathrm e^{-\lambda x}$ for every nonnegative $x$, hence $|\zeta-\eta|$ is exponential with parameter $\lambda$.
Edit Combining the second and third items here, one gets the result above.
