Laplace distribution

In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together along the abscissa, although the term is also sometimes used to refer to the Gumbel distribution. The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution.

Laplace

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \mu }$ location (real) ${\displaystyle b>0}$ scale (real)
Support	${\displaystyle \mathbb {R} }$
PDF	${\displaystyle {\frac {1}{2b}}\exp \left(-{\frac {|x-\mu |}{b}}\right)}$
CDF	${\displaystyle {\begin{cases}{\frac {1}{2}}\exp \left({\frac {x-\mu }{b}}\right)&{\text{if }}x\leq \mu \\[8pt]1-{\frac {1}{2}}\exp \left(-{\frac {x-\mu }{b}}\right)&{\text{if }}x\geq \mu \end{cases}}}$
Quantile	${\displaystyle {\begin{cases}\mu +b\ln \left(2F\right)&{\text{if }}F\leq {\frac {1}{2}}\\[8pt]\mu -b\ln \left(2-2F\right)&{\text{if }}F\geq {\frac {1}{2}}\end{cases}}}$
Mean	${\displaystyle \mu }$
Median	${\displaystyle \mu }$
Mode	${\displaystyle \mu }$
Variance	${\displaystyle 2b^{2}}$
MAD	${\displaystyle b\ln 2}$
Skewness	${\displaystyle 0}$
Ex. kurtosis	${\displaystyle 3}$
Entropy	${\displaystyle \ln(2be)}$
MGF	${\displaystyle {\frac {\exp(\mu t)}{1-b^{2}t^{2}}}{\text{ for }}|t|<1/b}$
CF	${\displaystyle {\frac {\exp(\mu it)}{1+b^{2}t^{2}}}}$
Expected shortfall	${\displaystyle {\begin{cases}\mu +b\left({\frac {p}{1-p}}\right)(1-\ln(2p))&,p<.5\\\mu +b\left(1-\ln \left(2(1-p)\right)\right)&,p\geq .5\end{cases}}}$