Laplace distribution

Laplace

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \mu }$ location (real) ${\displaystyle b>0}$ scale (real)
Support	${\displaystyle x\in (-\infty ;+\infty )\,}$
PDF	${\displaystyle {\frac {1}{2\,b}}\exp \left(-{\frac {|x-\mu |}{b}}\right)\,}$
CDF	${\displaystyle {\begin{cases}{\frac {1}{2}}\exp \left({\frac {x-\mu }{b}}\right)&{\mbox{if }}x<\mu \\[8pt]1-{\frac {1}{2}}\exp \left(-{\frac {x-\mu }{b}}\right)&{\mbox{if }}x\geq \mu \end{cases}}}$
Mean	${\displaystyle \mu }$
Median	${\displaystyle \mu }$
Mode	${\displaystyle \mu }$
Variance	${\displaystyle 2b^{2}}$
Skewness	${\displaystyle 0}$
Ex. kurtosis	${\displaystyle 3}$
Entropy	${\displaystyle \log(2be)}$
MGF	${\displaystyle {\frac {\exp(\mu \,t)}{1-b^{2}\,t^{2}}}\,\!{\text{ for }}|t|<1/b\,}$
CF	${\displaystyle {\frac {\exp(\mu \,i\,t)}{1+b^{2}\,t^{2}}}\,\!}$

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 back-to-back, although the term 'double exponential distribution' 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.

A random variable has a ${\displaystyle {\textrm {Laplace}}(\mu ,b)}$ distribution if its probability density function is

Here, ${\displaystyle \mu }$ is a location parameter and ${\displaystyle b>0}$, which is sometimes referred to as the diversity, is a scale parameter. If ${\displaystyle \mu =0}$ and ${\displaystyle b=1}$, the positive half-line is exactly an exponential distribution scaled by 1/2.