Editing Laplace distribution (section)

{{Short description|Probability distribution}}
{{Probability distribution
|name       =Laplace
|type       =density
|pdf_image  =[[Image:Laplace pdf mod.svg|325px|Probability density plots of Laplace distributions]]
|cdf_image  =[[Image:Laplace cdf mod.svg|325px|Cumulative distribution plots of Laplace distributions]]
|parameters =<math>\mu</math> [[location parameter|location]] ([[real number|real]])<br /><math>b > 0</math> [[scale parameter|scale]] (real)
|support    =<math>\mathbb{R}</math>
|pdf        =<math>\frac{1}{2b} \exp \left(-\frac{|x-\mu|}b \right)</math>
|cdf        =<math>\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}</math>

|quantile   =<math>\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}</math>

|mean       =<math>\mu</math>
|median     =<math>\mu</math>
|mode       =<math>\mu</math>
|variance   =<math>2b^2</math>
|mad        =<math>b \ln 2</math>
|skewness   =<math>0</math>
|kurtosis   =<math>3</math>
|entropy    =<math>\log(2be)</math>
|mgf        =<math>\frac{\exp(\mu t)}{1-b^2 t^2} \text{ for }|t| < 1/b</math>
|char       =<math>\frac{\exp(\mu it)}{1+b^2 t^2}</math>
|ES         =<math> \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}</math><ref name="norton">{{cite journal |last1=Norton |first1=Matthew |last2=Khokhlov |first2=Valentyn |last3=Uryasev |first3=Stan |year=2019 |title=Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation |journal=Annals of Operations Research |volume=299 |issue=1–2 |pages=1281–1315 |publisher=Springer|doi=10.1007/s10479-019-03373-1 |arxiv=1811.11301 |url=http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |access-date=2023-02-27}}</ref>
|bPOE       =<math>\begin{cases}
 \frac{1}{2}e^{1-\left(\frac{x-\mu}{b} \right)}&, x \geq \mu +b \\
1 - \frac{z}{\mathcal{W}( -2e^{-z-1}z)} &, x < \mu + b
\end{cases}</math>
Where <math>z=\frac{x-\mu}{b}</math>, and <math>\mathcal{W}</math> is the [[Lambert W function|Lambert-W function]]<ref name="norton"/>
}}
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 distribution]]s (with an additional location parameter) spliced together along the x-axis,<ref>{{Citation |last=Chattamvelli |first=Rajan |title=Laplace Distribution |date=2021 |work=Continuous Distributions in Engineering and the Applied Sciences – Part II |pages=189–199 |url=https://link.springer.com/10.1007/978-3-031-02435-1_4 |access-date=2025-04-04 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-031-02435-1_4 |isbn=978-3-031-01307-2 |last2=Shanmugam |first2=Ramalingam}}</ref> although the term is also sometimes used to refer to the [[Gumbel distribution]].  The difference between two [[Independent identically-distributed random variables|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|reason=Brownian motion is not mentioned elsewhere in this article. Either this topic should be expanded upon in the article, or there should be a citation justifying the claim.|date=October 2023}}.  Increments of [[Laplace motion]] or a [[variance gamma process]] evaluated over the time scale also have a Laplace distribution.