Log-Laplace distribution

Log-Laplace distribution
Log-Laplace distribution
	Probability density function Probability density functions for Log-Laplace distributions with varying parameters and .
	Cumulative distribution function Cumulative distribution functions for Log-Laplace distributions with varying parameters and .
Parameters	(position), ; (scale)
Support
PDF
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy

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.

\mu >0

(position),

b>0\,

(scale)

x\in (0,+\infty )

{\frac {1}{2bx}}\exp \left(-{\frac {|\ln x-\mu |}{b}}\right)

{\frac {e^{\mu }}{1-b^{2}}}

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Characterization

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	$x\in (0,+\infty )$
PDF	${\frac {1}{2bx}}\exp \left(-{\frac {\|\ln x-\mu \|}{b}}\right)$
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	${\frac {4b^{5}+14b^{3}+30b}{2-9b^{4}-17b^{2}}}{\sqrt {\frac {1-4b^{2}}{b^{2}+2}}}$
Excess kurtosis	${\frac {832b^{2}-3786b^{4}-2217b^{6}-241b^{8}+12}{144b^{8}+551b^{6}+477b^{4}-96b^{2}+4}}+b^{2}$
Entropy	$\ln \left(2b\right)+\mu +1$