Log-Laplace distribution

Short description: Probability distribution

Log-Laplace distribution

Probability density function Probability density functions for Log-Laplace distributions with varying parameters $\mu$ and $b$.
Cumulative distribution function Cumulative distribution functions for Log-Laplace distributions with varying parameters $\mu$ and $b$.
Parameters	$\mu >0$ (position), $b>0$ (scale)
Support	$x\in (0,+\mathrm{\infty })$
PDF	$\frac{1}{2bx}\exp (-\frac{|\ln x-\mu |}{b})$
Mean	$\frac{e^{\mu }}{1-b^2}$
Median	$e^{\mu }$
Mode	$e^{\mu }$
Variance	$\frac{e^{2\mu }{b}^4+2e^{2\mu }{b}^2}{1-4b^6+9b^4-6b^2}$
Skewness	$\frac{4b^5+14b^3+30b}{2-9b^4-17b^2}\sqrt{\frac{1-4b^2}{b^2+2}}$
Kurtosis	$\frac{832b^2-3786b^4-2217b^6-241b^8+12}{144b^8+551b^6+477b^4-96b^2+4}+b^2$
Entropy	$\ln \left(2b\right)+\mu +1$

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.

Characterization

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

$f(x|\mu ,b)=\frac{1}{2bx}{e}^{-\frac{|\ln x-\mu |}{b}}$

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

$F(y)=0.5[1+\mathrm{sgn}(\ln (y)-\mu )(1-e^{-\frac{|\ln (y)-\mu |}{b}})]$

Generalization

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

Properties

The mean or expected value of a log-Laplace distributed random variable X with a location parameter μ and a scale parameter b is given by

$E(X)=\frac{e^{\mu }}{1-b^2}$

The variance of X is given by

$Var(X)=\frac{e^{2\mu }{b}^4+2e^{2\mu }{b}^2}{1-4b^6+9b^4-6b^2}$