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	${\displaystyle 0}$
Excess kurtosis	${\displaystyle 3}$
Entropy	${\displaystyle \log(2be)}$
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	${\displaystyle {\begin{cases}\mu +b\left({\frac {p}{1-p}}\right)(1-\ln(2p))&,p<.5\\\mu +b\left(1-\ln \left(2(1-p)\right)\right)&,p\geq .5\end{cases}}}$[1]

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 x-axis,^[2] 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^{[citation needed]}. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution.