# Log-Laplace distribution

In probability theory and statistics, the log-Laplace distribution is the probability distribution of a random variable whose logarithm has a Laplace distribution. If X has a Laplace distribution with parameters μ and b, then Y = eX has a log-Laplace distribution. The distributional properties can be derived from the Laplace distribution.

## Characterization

### Probability density function

A random variable has a log-Laplace(μ, b) distribution if its probability density function is:[1]

${\displaystyle f(x|\mu ,b)={\frac {1}{2bx}}\exp \left(-{\frac {|\ln x-\mu |}{b}}\right)}$
${\displaystyle ={\frac {1}{2bx}}\left\{{\begin{matrix}\exp \left(-{\frac {\mu -\ln x}{b}}\right)&{\mbox{if }}\ln x<\mu \\[8pt]\exp \left(-{\frac {\ln x-\mu }{b}}\right)&{\mbox{if }}\ln x\geq \mu \end{matrix}}\right.}$

The cumulative distribution function for Y when y > 0, is

${\displaystyle F(y)=0.5\,[1+\operatorname {sgn}(\ln(y)-\mu )\,(1-\exp(-|\ln(y)-\mu |/b))].}$

Versions of the log-Laplace distribution based on an asymmetric Laplace distribution also exist.[2] Depending on the parameters, including asymmetry, the log-Laplace may or may not have a finite mean and a finite variance.[2]