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
Quantile
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
CF	${\displaystyle \frac{\exp (\mu it)}{1+b^2{t}^2}}$
Expected shortfall	[1]

Inprobability theory andstatistics, theLaplace distribution is a continuousprobability distribution named afterPierre-Simon Laplace. It is also sometimes called thedouble exponential distribution, because it can be thought of as twoexponential distributions (with an additional location parameter) spliced together along the x-axis,^[2] although the term is also sometimes used to refer to theGumbel distribution. The difference between twoindependent identically distributed exponential random variables is governed by a Laplace distribution, as is aBrownian motion evaluated at an exponentially distributed random time^{[citation needed]}. Increments ofLaplace motion or avariance gamma process evaluated over the time scale also have a Laplace distribution.

Arandom variable has a${\displaystyle \mathrm{Laplace}(\mu ,b)}$ distribution if itsprobability density function is

${\displaystyle f(x\mid \mu ,b)=\frac{1}{2b}\exp \left(-\frac{|x-\mu |}{b}\right),}$

where${\displaystyle \mu }$ is alocation parameter, and${\displaystyle b>0}$, which is sometimes referred to as the "diversity", is ascale parameter. If${\displaystyle \mu =0}$ and${\displaystyle b=1}$, the positive half-line is exactly anexponential distribution scaled by 1/2.^[3]

The probability density function of the Laplace distribution is also reminiscent of thenormal distribution; however, whereas the normal distribution is expressed in terms of the squared difference from the mean${\displaystyle \mu }$, the Laplace density is expressed in terms of theabsolute difference from the mean. Consequently, the Laplace distribution has fatter tails than the normal distribution. It is a special case of thegeneralized normal distribution and thehyperbolic 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 thelogistic distribution,hyperbolic secant distribution, and theChampernowne distribution.

The Laplace distribution is easy tointegrate (if one distinguishes two symmetric cases) due to the use of theabsolute value function. Itscumulative distribution function is as follows:

The inverse cumulative distribution function is given by

${\displaystyle F^{-1}(p)=\mu -b\mathrm{sgn}(p-0.5)\ln (1-2|p-0.5|).}$

${\displaystyle {\mu }_r^{\prime }=(\frac{1}{2})\sum _{k=0}^r[\frac{r!}{(r-k)!}{b}^k{\mu }^{(r-k)}\{1+(-1{)}^k\}].}$

• If${\displaystyle X\sim \text{Laplace}(\mu ,b)}$ then${\displaystyle kX+c\sim \text{Laplace}(k\mu +c,|k|b)}$.
• If${\displaystyle X\sim \text{Laplace}(0,1)}$ then${\displaystyle bX\sim \text{Laplace}(0,b)}$.
• If${\displaystyle X\sim \text{Laplace}(0,b)}$ then${\displaystyle \left|X\right|\sim \text{Exponential}\left(b^{-1}\right)}$ (exponential distribution).
• If${\displaystyle X,Y\sim \text{Exponential}(\lambda )}$ then${\displaystyle X-Y\sim \text{Laplace}\left(0,{\lambda }^{-1}\right)}$．
• If${\displaystyle X\sim \text{Laplace}(\mu ,b)}$ then${\displaystyle \left|X-\mu \right|\sim \text{Exponential}(b^{-1})}$.
• If${\displaystyle X\sim \text{Laplace}(\mu ,b)}$ then${\displaystyle X\sim \text{EPD}(\mu ,b,1)}$ (exponential power distribution).
• If${\displaystyle X_1,...,X_4\sim \text{N}(0,1)}$ (normal distribution) then${\displaystyle X_1{X}_2-X_3{X}_4\sim \text{Laplace}(0,1)}$ and${\displaystyle (X_1^2-X_2^2+X_3^2-X_4^2)/2\sim \text{Laplace}(0,1)}$.
• If${\displaystyle X_i\sim \text{Laplace}(\mu ,b)}$ then${\displaystyle \frac{{\displaystyle 2}}{b}\sum _{i=1}^n|X_i-\mu |\sim {\chi }^2(2n)}$ (chi-squared distribution).
• If${\displaystyle X,Y\sim \text{Laplace}(\mu ,b)}$ then${\displaystyle \frac{|X-\mu |}{|Y-\mu |}\sim \mathrm{F}(2,2)}$. (F-distribution)
• If${\displaystyle X,Y\sim \text{U}(0,1)}$ (uniform distribution) then${\displaystyle \log (X/Y)\sim \text{Laplace}(0,1)}$.
• If${\displaystyle X\sim \text{Exponential}(\lambda )}$ and${\displaystyle Y\sim \text{Bernoulli}(0.5)}$ (Bernoulli distribution) independent of${\displaystyle X}$, then${\displaystyle X(2Y-1)\sim \text{Laplace}\left(0,{\lambda }^{-1}\right)}$.
• If${\displaystyle X\sim \text{Exponential}(\lambda )}$ and${\displaystyle Y\sim \text{Exponential}(\nu )}$ independent of${\displaystyle X}$, then${\displaystyle \lambda X-\nu Y\sim \text{Laplace}(0,1)}$．
• If${\displaystyle X}$ has aRademacher distribution and${\displaystyle Y\sim \text{Exponential}(\lambda )}$ then${\displaystyle XY\sim \text{Laplace}(0,1/\lambda )}$.
• If${\displaystyle V\sim \text{Exponential}(1)}$ and${\displaystyle Z\sim N(0,1)}$ independent of${\displaystyle V}$, then${\displaystyle X=\mu +b\sqrt{2V}Z\sim \mathrm{L}\mathrm{a}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{e}(\mu ,b)}$.
• If${\displaystyle X\sim \text{GeometricStable}(2,0,\lambda ,0)}$ (geometric stable distribution) then${\displaystyle X\sim \text{Laplace}(0,\lambda )}$.
• The Laplace distribution is a limiting case of thehyperbolic distribution.
• If${\displaystyle X|Y\sim \text{N}(\mu ,Y^2)}$ with${\displaystyle Y\sim \text{Rayleigh}(b)}$ (Rayleigh distribution) then${\displaystyle X\sim \text{Laplace}(\mu ,b)}$. Note that if${\displaystyle Y\sim \text{Rayleigh}(b)}$, then${\displaystyle Y^2\sim \text{Gamma}(1,2b^2)}$ with${\displaystyle \text{E}(Y^2)=2b^2}$, which in turn equals the exponential distribution${\displaystyle \text{Exp}(1/(2b^2))}$.
• Given an integer${\displaystyle n\ge 1}$, if${\displaystyle X_i,Y_i\sim \mathrm{\Gamma }\left(\frac{1}{n},b\right)}$ (gamma distribution, using${\displaystyle k,\theta }$ characterization), then${\displaystyle \sum _{i=1}^n\left(\frac{\mu }{n}+X_i-Y_i\right)\sim \text{Laplace}(\mu ,b)}$ (infinite divisibility)^[4]
• IfX has a Laplace distribution, thenY =e^X has a log-Laplace distribution; conversely, ifX has a log-Laplace distribution, then itslogarithm has a Laplace distribution.

Let${\displaystyle X,Y}$ be independent laplace random variables:${\displaystyle X\sim \text{Laplace}({\mu }_X,b_X)}$ and${\displaystyle Y\sim \text{Laplace}({\mu }_Y,b_Y)}$, and we want to compute${\displaystyle P(X>Y)}$.

The probability of${\displaystyle P(X>Y)}$ can be reduced (using the properties below) to${\displaystyle P(\mu +b{Z}_1>Z_2)}$, where${\displaystyle Z_1,Z_2\sim \text{Laplace}(0,1)}$. This probability is equal to

When${\displaystyle b=1}$, both expressions are replaced by their limit as${\displaystyle b\to 1}$:

To compute the case for${\displaystyle \mu >0}$, note that

since${\displaystyle Z\sim -Z}$ when${\displaystyle Z\sim \text{Laplace}(0,1)}$ .

A Laplace random variable can be represented as the difference of twoindependent and identically distributed (iid) exponential random variables.^[4] One way to show this is by using thecharacteristic 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${\displaystyle X,Y\sim \text{Exponential}(\lambda )}$. The characteristic functions for${\displaystyle X,-Y}$ are

${\displaystyle \frac{\lambda }{-it+\lambda },\frac{\lambda }{it+\lambda }}$

respectively. On multiplying these characteristic functions (equivalent to the characteristic function of the sum of the random variables${\displaystyle X+(-Y)}$), the result is

${\displaystyle \frac{{\lambda }^2}{(-it+\lambda )(it+\lambda )}=\frac{{\lambda }^2}{t^2+{\lambda }^2}.}$

This is the same as the characteristic function for${\displaystyle Z\sim \text{Laplace}(0,1/\lambda )}$, which is

${\displaystyle \frac{1}{1+\frac{t^2}{{\lambda }^2}}.}$

Sargan distributions are a system of distributions of which the Laplace distribution is a core member. A${\displaystyle p}$th order Sargan distribution has density^[5][6]

${\displaystyle f_p(x)=\frac{1}{2}\exp (-\alpha |x|)\frac{{\displaystyle 1+\sum _{j=1}^p{\beta }_j{\alpha }^j|x{|}^j}}{{\displaystyle 1+\sum _{j=1}^{p}j!{\beta }_j}},}$

for parameters${\displaystyle \alpha \ge 0,{\beta }_j\ge 0}$. The Laplace distribution results for${\displaystyle p=0}$.

Given${\displaystyle n}$ independent and identically distributed samples${\displaystyle x_1,x_2,...,x_n}$, themaximum likelihood (MLE) estimator of${\displaystyle \mu }$ is the samplemedian,^[7]

${\displaystyle \hat{\mu }=\mathrm{m}\mathrm{e}\mathrm{d}(x).}$

The MLE estimator of${\displaystyle b}$ is themean absolute deviation from the median,^{[citation needed]}

${\displaystyle \hat{b}=\frac{1}{n}\sum _{i=1}^n|x_i-\hat{\mu }|.}$

revealing a link between the Laplace distribution andleast absolute deviations.A correction for small samples can be applied as follows:

${\displaystyle {\hat{b}}^{\ast }=\hat{b}\cdot n/(n-2)}$

(see:exponential distribution#Parameter estimation).

The Laplacian distribution has been used in speech recognition to model priors onDFT coefficients^[8] and in JPEG image compression to model AC coefficients^[9] generated by aDCT.

• The addition of noise drawn from a Laplacian distribution, with scaling parameter appropriate to a function's sensitivity, to the output of astatistical database query is the most common means to providedifferential privacy in statistical databases.

• Inregression analysis, theleast absolute deviations estimate arises as the maximum likelihood estimate if the errors have a Laplace distribution.
• TheLasso can be thought of as aBayesian regression with a Laplacianprior for the coefficients.^[11]
• Inhydrology the Laplace distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture, made withCumFreq, illustrates an example of fitting the Laplace distribution to ranked annually maximum one-day rainfalls showing also the 90%confidence belt based on thebinomial distribution. The rainfall data are represented byplotting positions as part of thecumulative 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 anasymmetric Laplace distribution) to address problems ofskewness,kurtosis and thevolatility smile that often occur when using a normal distribution for pricing these instruments.^[12][13]

The Laplace distribution, being acomposite ordouble 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.^[14]

Given a random variable${\displaystyle U}$ drawn from theuniform distribution in the interval${\displaystyle \left(-1/2,1/2\right)}$, the random variable

${\displaystyle X=\mu -b\mathrm{sgn}(U)\ln (1-2|U|)}$

has a Laplace distribution with parameters${\displaystyle \mu }$ and${\displaystyle b}$. This follows from the inverse cumulative distribution function given above.

A${\displaystyle \text{Laplace}(0,b)}$variate can also be generated as the difference of twoi.i.d.${\displaystyle \text{Exponential}(1/b)}$ random variables. Equivalently,${\displaystyle \text{Laplace}(0,1)}$ can also be generated as thelogarithm of the ratio of twoi.i.d. uniform random variables.

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 thecentral limit theorem.^[15][16]

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.^[17]

• Generalized normal distribution#Symmetric version
• Multivariate Laplace distribution
• Besov measure, a generalisation of the Laplace distribution tofunction spaces
• Cauchy distribution, also called the "Lorentzian distribution", ie the Fourier transform of the Laplace
• Characteristic function (probability theory)

• "Laplace distribution",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

• Continuous distributions
• Compound probability distributions
• Pierre-Simon Laplace
• Exponential family distributions
• Location-scale family probability distributions
• Geometric stable distributions
• Infinitely divisible probability distributions

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