Editing Laplace distribution

{{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.

==Definitions==

===Probability density function===
A [[random variable]] has a <math>\operatorname{Laplace}(\mu, b)</math> distribution if its [[probability density function]] is

: <math>f(x \mid \mu, b) = \frac{1}{2b} \exp\left( -\frac{|x - \mu|}{b} \right),</math>

where <math>\mu</math> is a [[location parameter]], and <math>b > 0</math>, which is sometimes referred to as the "diversity", is a [[scale parameter]]. If <math>\mu = 0</math> and <math>b = 1</math>, the positive half-line is exactly an [[exponential distribution]] scaled by 1/2.<ref>{{cite journal | last1 = Huang | first1 = Yunfei. | display-authors = etal | year = 2022 | title = Sparse inference and active learning of stochastic differential equations from data | journal = Scientific Reports | volume = 12 | number = 1| page = 21691 | doi = 10.1038/s41598-022-25638-9  | pmid = 36522347 | doi-access = free| pmc = 9755218 | arxiv = 2203.11010 | bibcode = 2022NatSR..1221691H }}</ref>

The probability density function of the Laplace distribution is also reminiscent of the [[normal distribution]]; however, whereas the normal distribution is expressed in terms of the squared difference from the mean <math>\mu</math>, the Laplace density is expressed in terms of the [[absolute difference]] from the mean. Consequently, the Laplace distribution has fatter tails than the normal distribution. It is a special case of the [[generalized normal distribution]] and the [[hyperbolic distribution]]. Continuous symmetric distributions that have exponential tails, like the Laplace distribution, but which have probability density functions that are differentiable at the mode include the [[logistic distribution]], [[hyperbolic secant distribution]], and the [[Champernowne distribution]].

===Cumulative distribution function===
The Laplace distribution is easy to [[integral|integrate]] (if one distinguishes two symmetric cases) due to the use of the [[absolute value]] function.  Its [[cumulative distribution function]] is as follows:
:<math>\begin{align}
F(x) &= \int_{-\infty}^x \!\!f(u)\,\mathrm{d}u  = \begin{cases}
             \frac12 \exp \left( \frac{x-\mu}{b} \right) & \mbox{if }x < \mu \\
             1-\frac12 \exp \left( -\frac{x-\mu}{b} \right) & \mbox{if }x \geq \mu
            \end{cases} \\
&=\tfrac{1}{2} + \tfrac{1}{2} \sgn(x-\mu) \left(1-\exp \left(-\frac{|x-\mu|}{b} \right ) \right ).
\end{align}</math>

The inverse cumulative distribution function is given by

:<math>F^{-1}(p) = \mu - b\,\sgn(p-0.5)\,\ln(1 - 2|p-0.5|).</math>

==Properties==
===Moments===

:<math>\mu_r' = \bigg({\frac{1}{2}}\bigg) \sum_{k=0}^r \bigg[{\frac{r!}{(r-k)!}} b^k \mu^{(r-k)} \{1 + (-1)^k\}\bigg].</math>

===Related distributions===
*If <math>X \sim \textrm{Laplace}(\mu, b)</math> then <math>kX + c \sim \textrm{Laplace}(k\mu + c, |k|b)</math>.
*If <math>X \sim \textrm{Laplace}(0, 1)</math> then <math>bX \sim \textrm{Laplace}(0, b)</math>.
*If <math>X \sim \textrm{Laplace}(0, b)</math> then <math>\left|X\right| \sim \textrm{Exponential}\left(b^{-1}\right)</math> ([[exponential distribution]]).
*If <math>X, Y \sim \textrm{Exponential}(\lambda)</math> then <math>X - Y \sim \textrm{Laplace}\left(0, \lambda^{-1}\right)</math>．
*If <math>X \sim \textrm{Laplace}(\mu, b)</math> then <math>\left|X - \mu\right| \sim \textrm{Exponential}(b^{-1})</math>.
*If <math>X \sim \textrm{Laplace}(\mu, b)</math> then <math>X \sim \textrm{EPD}(\mu, b, 1)</math> ([[exponential power distribution]]).
*If <math>X_1, ...,X_4 \sim \textrm{N}(0, 1)</math> ([[normal distribution]]) then <math>X_1X_2 - X_3X_4 \sim \textrm{Laplace}(0, 1)</math> and <math>(X_1^2 - X_2^2 + X_3^2 - X_4^2)/2 \sim \textrm{Laplace}(0, 1)</math>.
*If <math>X_i \sim \textrm{Laplace}(\mu, b)</math> then <math>\frac{\displaystyle 2}{b} \sum_{i=1}^n |X_i-\mu| \sim \chi^2(2n)</math> ([[chi-squared distribution]]).
*If <math>X, Y \sim \textrm{Laplace}(\mu, b)</math> then <math>\tfrac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2)</math>. ([[F-distribution]])
*If <math>X, Y \sim \textrm{U}(0, 1)</math> ([[Uniform distribution (continuous)|uniform distribution]]) then <math>\log(X/Y) \sim \textrm{Laplace}(0, 1)</math>.
*If <math>X \sim \textrm{Exponential}(\lambda)</math> and <math>Y \sim \textrm{Bernoulli}(0.5)</math> ([[Bernoulli distribution]]) independent of <math>X</math>, then <math>X(2Y - 1) \sim \textrm{Laplace}\left(0, \lambda^{-1}\right)</math>.
*If <math>X \sim \textrm{Exponential}(\lambda)</math> and <math>Y \sim \textrm{Exponential}(\nu)</math> independent of <math>X</math>, then <math>\lambda X - \nu Y \sim \textrm{Laplace}(0, 1)</math>．
*If <math>X</math> has a [[Rademacher distribution]] and <math>Y \sim \textrm{Exponential}(\lambda)</math> then <math>XY \sim \textrm{Laplace}(0, 1/\lambda)</math>.
*If <math>V \sim \textrm{Exponential}(1)</math> and <math>Z \sim N(0, 1)</math> independent of <math>V</math>, then <math>X = \mu + b \sqrt{2 V}Z \sim \mathrm{Laplace}(\mu,b)</math>.
*If <math>X \sim \textrm{GeometricStable}(2, 0, \lambda, 0)</math> ([[geometric stable distribution]]) then <math>X \sim \textrm{Laplace}(0, \lambda)</math>.
*The Laplace distribution is a limiting case of the [[hyperbolic distribution]].
*If <math>X|Y \sim \textrm{N}(\mu,Y^2)</math> with <math>Y \sim \textrm{Rayleigh}(b)</math> ([[Rayleigh distribution]]) then <math>X \sim \textrm{Laplace}(\mu, b)</math>. Note that if <math>Y \sim \textrm{Rayleigh}(b)</math>, then <math>Y^2 \sim \textrm{Gamma}(1,2b^2) </math> with <math>\textrm{E}(Y^2)=2b^2</math>, which in turn equals the exponential distribution <math>\textrm{Exp}(1/(2b^2))</math>.
*Given an integer <math>n \ge 1</math>, if <math>X_i, Y_i \sim \Gamma\left(\frac{1}{n}, b\right)</math> ([[gamma distribution]], using <math>k, \theta</math> characterization), then <math>\sum_{i=1}^n \left( \frac{\mu}{n} + X_i - Y_i\right) \sim \textrm{Laplace}(\mu, b)</math> ([[Infinite divisibility (probability)|infinite divisibility]])<ref name="Kotz">{{cite book |title=The Laplace distribution and generalizations: a revisit with applications to Communications, Economics, Engineering and Finance |first1= Samuel|last1= Kotz |first2=Tomasz J. |last2=Kozubowski |first3=Krzysztof |last3= Podgórski |pages=23 (Proposition 2.2.2, Equation 2.2.8) |url=https://books.google.com/books?id=cb8B07hwULUC&q=laplace+distribution+exponential+characteristic+function&pg=PA23|isbn= 9780817641665 |publisher=Birkhauser|year=2001}}</ref>
* If ''X'' has a Laplace distribution, then ''Y'' = ''e''<sup>''X''</sup> has a log-Laplace distribution; conversely, if ''X'' has a log-Laplace distribution, then its [[logarithm]] has a Laplace distribution.

===Probability of a Laplace being greater than another ===

Let  <math>X, Y</math> be independent laplace random variables: <math>X \sim \textrm{Laplace}(\mu_X, b_X)</math> and <math>Y \sim \textrm{Laplace}(\mu_Y, b_Y)</math>, and we want to compute <math>P(X>Y)</math>.

The probability of <math>P(X > Y)</math> can be reduced (using the properties below) to <math>P(\mu + bZ_1 > Z_2)</math>, where <math>Z_1, Z_2 \sim  \textrm{Laplace}(0, 1)</math>. This probability is equal to

<math>P(\mu + bZ_1 > Z_2) = \begin{cases}
\frac{b^2 e^{\mu/b} - e^\mu}{2(b^2 -1)}, & \text{when } \mu < 0 \\
1-\frac{ b^2 e^{-\mu/b} - e^{-\mu}}{2(b^2 -1)}, & \text{when } \mu > 0 \\
\end{cases}</math>

When <math> b = 1 </math>, both expressions are replaced by their limit as <math> b \to 1</math>:

<math>P(\mu + Z_1 > Z_2) = \begin{cases}
e^\mu\frac{(2-\mu)}4, & \text{when } \mu < 0 \\
1-e^{-\mu}\frac{(2+\mu)}4, & \text{when } \mu > 0 \\
\end{cases}</math>

To compute the case for <math> \mu > 0 </math>, note that <math>P(\mu + Z_1 > Z_2) = 1 - P(\mu + Z_1 < Z_2) = 1 - P(-\mu - Z_1 > -Z_2) = 1 - P(-\mu + Z_1 > Z_2)</math>

since <math> Z \sim -Z </math> when <math>Z \sim \textrm{Laplace}(0, 1) </math> .

===Relation to the exponential distribution===
A Laplace random variable can be represented as the difference of two [[independent and identically distributed]] ([[iid]]) exponential random variables.<ref name="Kotz" /> One way to show this is by using the [[characteristic function (probability theory)|characteristic function]] approach. For any set of independent continuous random variables, for any linear combination of those variables, its characteristic function (which uniquely determines the distribution) can be acquired by multiplying the corresponding characteristic functions.

Consider two i.i.d random variables <math>X, Y \sim \textrm{Exponential}(\lambda)</math>. The characteristic functions for <math>X, -Y</math> are

:<math>\frac{\lambda }{-i t+\lambda }, \quad \frac{\lambda }{i t+\lambda }</math>

respectively. On multiplying these characteristic functions (equivalent to the characteristic function of the sum of the random variables <math>X + (-Y)</math>), the result is

:<math>\frac{\lambda ^2}{(-i t+\lambda ) (i t+\lambda )} = \frac{\lambda ^2}{t^2+\lambda ^2}.</math>

This is the same as the characteristic function for <math>Z \sim \textrm{Laplace}(0,1/\lambda)</math>, which is

:<math>\frac{1}{1+\frac{t^2}{\lambda ^2}}.</math>

===Sargan distributions===
Sargan distributions are a system of distributions of which the Laplace distribution is a core member. A <math>p</math>th order Sargan distribution has density<ref>Everitt, B.S. (2002) ''The Cambridge Dictionary of Statistics'', CUP. {{isbn|0-521-81099-X}}</ref><ref>Johnson, N.L., Kotz S., Balakrishnan, N. (1994) ''Continuous Univariate Distributions'', Wiley. {{isbn|0-471-58495-9}}. p. 60</ref>
:<math>f_p(x)=\tfrac{1}{2} \exp(-\alpha |x|)  \frac{\displaystyle 1+\sum_{j=1}^p \beta_j \alpha^j |x|^j}{\displaystyle 1+\sum_{j=1}^p j!\beta_j},</math>
for parameters <math>\alpha \ge 0, \beta_j \ge 0</math>. The Laplace distribution results for <math>p = 0</math>.

==Statistical inference==
Given <math>n</math> independent and identically distributed samples <math>x_1, x_2, ..., x_n</math>, the [[maximum likelihood]] (MLE) estimator of <math>\mu</math> is the sample [[median]],<ref>{{Cite journal
 | author = Robert M. Norton
 | author-link = Robert M. Norton
 | title = The Double Exponential Distribution: Using Calculus to Find a Maximum Likelihood Estimator
 | journal = [[The American Statistician]]
 | volume = 38
 | issue = 2
 |date=May 1984
 | pages = 135&ndash;136
 | doi = 10.2307/2683252
 | jstor = 2683252
 | publisher = American Statistical Association
}}</ref>
:<math>\hat{\mu} = \mathrm{med}(x).</math>

The MLE estimator of <math>b</math> is the [[mean absolute deviation]] from the median,{{citation needed|date=August 2022}}
:<math>\hat{b} = \frac{1}{n} \sum_{i = 1}^{n} |x_i - \hat{\mu}|.</math>
revealing a link between the Laplace distribution and [[least absolute deviations]].
A correction for small samples can be applied as follows:
:<math>\hat{b}^* = \hat{b} \cdot n/(n-2)</math>
(see: [[exponential distribution#Parameter estimation]]).

==Occurrence and applications==
The Laplacian distribution has been used in speech recognition to model priors on [[Discrete Fourier transform|DFT]] coefficients <ref>{{Cite journal 
| last1 = Eltoft 
| first1 = T. 
| last2 = Taesu Kim 
| last3 = Te-Won Lee 
| doi = 10.1109/LSP.2006.870353 
| title = On the multivariate Laplace distribution 
| journal = IEEE Signal Processing Letters 
| url = http://eo.uit.no/publications/TE-SPL-06.pdf 
| volume = 13 
| issue = 5 
| pages = 300–303 
| year = 2006 
| bibcode = 2006ISPL...13..300E 
| s2cid = 1011487 
| access-date = 2012-07-04 
| archive-url = https://web.archive.org/web/20130606114728/http://eo.uit.no/publications/TE-SPL-06.pdf 
| archive-date = 2013-06-06 
| url-status = dead 
}}</ref> and in JPEG image compression to model AC coefficients <ref>{{Cite journal
| last1 = Minguillon | first1 = J.
| last2 = Pujol | first2 = J.
| doi = 10.1117/1.1344592
| title = JPEG standard uniform quantization error modeling with applications to sequential and progressive operation modes
| journal = Journal of Electronic Imaging
| volume = 10
| issue = 2
| pages = 475–485
| year = 2001 | bibcode = 2001JEI....10..475M
| hdl = 10609/6263
| url = http://openaccess.uoc.edu/webapps/o2/bitstream/10609/6263/6/jei-jpeg.pdf| hdl-access = free
}}</ref> generated by a [[Discrete cosine transform|DCT]].

*The addition of noise drawn from a Laplacian distribution, with scaling parameter appropriate to a function's sensitivity, to the output of a [[statistical database]] query is the most common means to provide [[differential privacy]] in statistical databases.
[[File:Laplace Surinam.png|thumb|300px|Fitted Laplace distribution to maximum one-day rainfalls <ref>[https://www.waterlog.info/cumfreq.htm CumFreq for probability distribution fitting]</ref> ]]

*In [[regression analysis]], the [[least absolute deviations]] estimate arises as the maximum likelihood estimate if the errors have a Laplace distribution.
*The [[Lasso (statistics)|Lasso]] can be thought of as a [[Bayesian regression]] with a Laplacian [[prior distribution|prior]] for the coefficients.<ref>{{cite book |first=Scott |last=Pardo |title=Statistical Analysis of Empirical Data Methods for Applied Sciences |publisher=Springer |year=2020 |isbn=978-3-030-43327-7 |page=58 |url=https://books.google.com/books?id=k0nhDwAAQBAJ&pg=PA58 }}</ref>
* In [[hydrology]] the Laplace distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture, made with [[CumFreq]], illustrates an example of fitting the Laplace distribution to ranked annually maximum one-day rainfalls showing also the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by [[plotting position]]s as part of the [[cumulative frequency analysis]].
* The Laplace distribution has applications in finance.  For example, S.G. Kou developed a model for financial instrument prices incorporating a Laplace distribution (in some cases an [[asymmetric Laplace distribution]]) to address problems of [[skewness]], [[kurtosis]] and the [[volatility smile]] that often occur when using a normal distribution for pricing these instruments.<ref>{{cite journal|title=A Jump-Diffusion Model for Option Pricing|author=Kou, S.G.|url=https://www.jstor.org/stable/822677|date=August 8, 2002|accessdate=2022-03-01|journal=Management Science|volume=48|issue=8|pages=1086–1101|doi=10.1287/mnsc.48.8.1086.166 |jstor=822677 }}</ref><ref>{{cite book|title=General Equilibrium Option Pricing Method: Theoretical and Empirical Study|page=70|author=Chen, Jian|year=2018|publisher=Springer|isbn=9789811074288}}</ref>

: The Laplace distribution, being a '''composite''' or '''double''' distribution, is applicable in situations where the lower values originate under different external conditions than the higher ones so that they follow a different pattern.<ref>[https://www.waterlog.info/composite.htm A collection of composite distributions]</ref>

==Random variate generation==
{{further|Non-uniform random variate generation}}

Given a random variable <math>U</math> drawn from the [[uniform distribution (continuous)|uniform distribution]] in the interval <math>\left(-1/2, 1/2\right)</math>, the random variable

:<math>X=\mu - b\,\sgn(U)\,\ln(1 - 2|U|)</math>

has a Laplace distribution with parameters <math>\mu</math> and <math>b</math>. This follows from the inverse cumulative distribution function given above.

A <math>\textrm{Laplace}(0, b)</math> [[variate]] can also be generated as the difference of two [[Independent identically-distributed random variables|i.i.d.]] <math>\textrm{Exponential}(1/b)</math> random variables. Equivalently, <math>\textrm{Laplace}(0,1)</math> can also be generated as the [[logarithm]] of the ratio of two [[Independent identically-distributed random variables|i.i.d.]] uniform random variables.

==History==
This distribution is often referred to as "Laplace's first law of errors". He published it in 1774, modeling the frequency of an error as an exponential function of its magnitude once its sign was disregarded. Laplace would later replace this model with his "second law of errors", based on the normal distribution, after the discovery of the [[central limit theorem]].<ref name=Laplace1774>Laplace, P-S. (1774). Mémoire sur la probabilité des causes par les évènements. Mémoires de l’Academie Royale des Sciences Presentés par Divers Savan, 6, 621–656</ref><ref name=Wilson1923>{{cite journal | last=Wilson | first=Edwin Bidwell | title=First and Second Laws of Error | journal=Journal of the American Statistical Association | publisher=Informa UK Limited | volume=18 | issue=143 | year=1923 | issn=0162-1459 | doi=10.1080/01621459.1923.10502116 | pages=841–851}} {{PD-notice}}</ref>

[[Keynes]] published a paper in 1911 based on his earlier thesis wherein he showed that the Laplace distribution minimised the absolute deviation from the median.<ref name=Keynes1911>{{cite journal | last=Keynes | first=J. M. | title=The Principal Averages and the Laws of Error which Lead to Them | journal=Journal of the Royal Statistical Society | publisher=JSTOR | volume=74 | issue=3 | year=1911 | pages=322–331 | issn=0952-8385 | doi=10.2307/2340444 | jstor=2340444 | url=https://zenodo.org/record/2253549 }}</ref>

==See also==
* [[Generalized normal distribution#Symmetric version]]
* [[Multivariate Laplace distribution]]
* [[Besov measure]], a generalisation of the Laplace distribution to [[function space]]s
* [[Cauchy distribution]], also called the "Lorentzian distribution", ie the Fourier transform of the Laplace
* [[Characteristic function (probability theory)]]

==References==
{{Reflist|30em}}

==External links==
* {{springer|title=Laplace distribution|id=p/l057460}}

{{ProbDistributions|continuous-infinite}}

{{DEFAULTSORT:Laplace Distribution}}
[[Category:Continuous distributions]]
[[Category:Compound probability distributions]]
[[Category:Pierre-Simon Laplace]]
[[Category:Exponential family distributions]]
[[Category:Location-scale family probability distributions]]
[[Category:Geometric stable distributions]]
[[Category:Infinitely divisible probability distributions]]