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 = e^X has a log-Laplace distribution. The distributional properties can be derived from the Laplace distribution.

Log-Laplace distribution

Probability density function Probability density functions for Log-Laplace distributions with varying parameters ${\displaystyle \mu }$ and ${\displaystyle b}$.

Cumulative distribution function Cumulative distribution functions for Log-Laplace distributions with varying parameters ${\displaystyle \mu }$ and ${\displaystyle b}$.

Characterization

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

${\displaystyle f(x|\mu ,b)={\frac {1}{2bx}}\exp \left(-{\frac {|\ln x-\mu |}{b}}\right)}$

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

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

Generalization

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]