Laplace distribution#Random variate generation

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

: $f(x\; \backslash mid\; \backslash mu,\; b)\; =\; \backslash frac\{1\}\{2b\}\; \backslash exp\backslash left(\; -\backslash frac$

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

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 $\backslash mu$, 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.

The Laplace distribution is easy to integrate (if one distinguishes two symmetric cases) due to the use of the absolute value function. Its cumulative distribution function is as follows:

:$\backslash begin\{align\}$

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

\end{align}

The inverse cumulative distribution function is given by

:$F^\{-1\}(p)\; =\; \backslash mu\; -\; b\backslash ,\backslash sgn(p-0.5)\backslash ,\backslash ln(1\; -\; 2|p-0.5|).$

:$\backslash mu\_r\text{'}\; =\; \backslash bigg(\{\backslash frac\{1\}\{2\}\}\backslash bigg)\; \backslash sum\_\{k=0\}^r\; \backslash bigg[\{\backslash frac\{r!\}\{(r-k)!\}\}\; b^k\; \backslash mu^\{(r-k)\}\; \backslash \{1\; +\; (-1)^k\backslash \}\backslash bigg].$

• If $X\; \backslash sim\; \backslash textrm\{Laplace\}(\backslash mu,\; b)$ then $kX\; +\; c\; \backslash sim\; \backslash textrm\{Laplace\}(k\backslash mu\; +\; c,\; |k|b)$.
• If $X\; \backslash sim\; \backslash textrm\{Laplace\}(0,\; 1)$ then $bX\; \backslash sim\; \backslash textrm\{Laplace\}(0,\; b)$.
• If $X\; \backslash sim\; \backslash textrm\{Laplace\}(0,\; b)$ then $\backslash left|X\backslash right|\; \backslash sim\; \backslash textrm\{Exponential\}\backslash left(b^\{-1\}\backslash right)$ (exponential distribution).
• If $X,\; Y\; \backslash sim\; \backslash textrm\{Exponential\}(\backslash lambda)$ then $X\; -\; Y\; \backslash sim\; \backslash textrm\{Laplace\}\backslash left(0,\; \backslash lambda^\{-1\}\backslash right)$．
• If $X\; \backslash sim\; \backslash textrm\{Laplace\}(\backslash mu,\; b)$ then $\backslash left|X\; -\; \backslash mu\backslash right|\; \backslash sim\; \backslash textrm\{Exponential\}(b^\{-1\})$.
• If $X\; \backslash sim\; \backslash textrm\{Laplace\}(\backslash mu,\; b)$ then $X\; \backslash sim\; \backslash textrm\{EPD\}(\backslash mu,\; b,\; 1)$ (exponential power distribution).
• If $X\_1,\; ...,X\_4\; \backslash sim\; \backslash textrm\{N\}(0,\; 1)$ (normal distribution) then $X\_1X\_2\; -\; X\_3X\_4\; \backslash sim\; \backslash textrm\{Laplace\}(0,\; 1)$ and $(X\_1^2\; -\; X\_2^2\; +\; X\_3^2\; -\; X\_4^2)/2\; \backslash sim\; \backslash textrm\{Laplace\}(0,\; 1)$.
• If $X\_i\; \backslash sim\; \backslash textrm\{Laplace\}(\backslash mu,\; b)$ then $\backslash frac\{\backslash displaystyle\; 2\}\{b\}\; \backslash sum\_\{i=1\}^n\; |X\_i-\backslash mu|\; \backslash sim\; \backslash chi^2(2n)$ (chi-squared distribution).
• If $X,\; Y\; \backslash sim\; \backslash textrm\{Laplace\}(\backslash mu,\; b)$ then $\backslash tfrac$

  X-\mu

  Y-\mu

  \sim \operatorname{F}(2,2). (F-distribution)
• If $X,\; Y\; \backslash sim\; \backslash textrm\{U\}(0,\; 1)$ (uniform distribution) then $\backslash log(X/Y)\; \backslash sim\; \backslash textrm\{Laplace\}(0,\; 1)$.
• If $X\; \backslash sim\; \backslash textrm\{Exponential\}(\backslash lambda)$ and $Y\; \backslash sim\; \backslash textrm\{Bernoulli\}(0.5)$ (Bernoulli distribution) independent of $X$, then $X(2Y\; -\; 1)\; \backslash sim\; \backslash textrm\{Laplace\}\backslash left(0,\; \backslash lambda^\{-1\}\backslash right)$.
• If $X\; \backslash sim\; \backslash textrm\{Exponential\}(\backslash lambda)$ and $Y\; \backslash sim\; \backslash textrm\{Exponential\}(\backslash nu)$ independent of $X$, then $\backslash lambda\; X\; -\; \backslash nu\; Y\; \backslash sim\; \backslash textrm\{Laplace\}(0,\; 1)$．
• If $X$ has a Rademacher distribution and $Y\; \backslash sim\; \backslash textrm\{Exponential\}(\backslash lambda)$ then $XY\; \backslash sim\; \backslash textrm\{Laplace\}(0,\; 1/\backslash lambda)$.
• If $V\; \backslash sim\; \backslash textrm\{Exponential\}(1)$ and $Z\; \backslash sim\; N(0,\; 1)$ independent of $V$, then $X\; =\; \backslash mu\; +\; b\; \backslash sqrt\{2\; V\}Z\; \backslash sim\; \backslash mathrm\{Laplace\}(\backslash mu,b)$.
• If $X\; \backslash sim\; \backslash textrm\{GeometricStable\}(2,\; 0,\; \backslash lambda,\; 0)$ (geometric stable distribution) then $X\; \backslash sim\; \backslash textrm\{Laplace\}(0,\; \backslash lambda)$.
• The Laplace distribution is a limiting case of the hyperbolic distribution.
• If $X|Y\; \backslash sim\; \backslash textrm\{N\}(\backslash mu,Y^2)$ with $Y\; \backslash sim\; \backslash textrm\{Rayleigh\}(b)$ (Rayleigh distribution) then $X\; \backslash sim\; \backslash textrm\{Laplace\}(\backslash mu,\; b)$. Note that if $Y\; \backslash sim\; \backslash textrm\{Rayleigh\}(b)$, then $Y^2\; \backslash sim\; \backslash textrm\{Gamma\}(1,2b^2)$ with $\backslash textrm\{E\}(Y^2)=2b^2$, which in turn equals the exponential distribution $\backslash textrm\{Exp\}(1/(2b^2))$.
• Given an integer $n\; \backslash ge\; 1$, if $X\_i,\; Y\_i\; \backslash sim\; \backslash Gamma\backslash left(\backslash frac\{1\}\{n\},\; b\backslash right)$ (gamma distribution, using $k,\; \backslash theta$ characterization), then $\backslash sum\_\{i=1\}^n\; \backslash left(\; \backslash frac\{\backslash mu\}\{n\}\; +\; X\_i\; -\; Y\_i\backslash right)\; \backslash sim\; \backslash textrm\{Laplace\}(\backslash mu,\; b)$ (infinite divisibility){{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}}
• If X has a Laplace distribution, then Y = e^X has a log-Laplace distribution; conversely, if X has a log-Laplace distribution, then its logarithm has a Laplace distribution.

Let $X,\; Y$ be independent laplace random variables: $X\; \backslash sim\; \backslash textrm\{Laplace\}(\backslash mu\_X,\; b\_X)$ and $Y\; \backslash sim\; \backslash textrm\{Laplace\}(\backslash mu\_Y,\; b\_Y)$, and we want to compute $P(X>Y)$.

The probability of $P(X\; >\; Y)$ can be reduced (using the properties below) to $P(\backslash mu\; +\; bZ\_1\; >\; Z\_2)$, where $Z\_1,\; Z\_2\; \backslash sim\; \backslash textrm\{Laplace\}(0,\; 1)$. This probability is equal to

$P(\backslash mu\; +\; bZ\_1\; >\; Z\_2)\; =\; \backslash begin\{cases\}$

\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}

When $b\; =\; 1$, both expressions are replaced by their limit as $b\; \backslash to\; 1$:

$P(\backslash mu\; +\; Z\_1\; >\; Z\_2)\; =\; \backslash begin\{cases\}$

e^\mu\frac{(2-\mu)}4, & \text{when } \mu < 0 \\

1-e^{-\mu}\frac{(2+\mu)}4, & \text{when } \mu > 0 \\

\end{cases}

To compute the case for $\backslash mu\; >\; 0$, note that

since $Z\; \backslash sim\; -Z$ when $Z\; \backslash sim\; \backslash textrm\{Laplace\}(0,\; 1)$ .

A Laplace random variable can be represented as the difference of two independent and identically distributed (iid) exponential random variables. One way to show this is by using the 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 $X,\; Y\; \backslash sim\; \backslash textrm\{Exponential\}(\backslash lambda)$. The characteristic functions for $X,\; -Y$ are

:$\backslash frac\{\backslash lambda\; \}\{-i\; t+\backslash lambda\; \},\; \backslash quad\; \backslash frac\{\backslash lambda\; \}\{i\; t+\backslash lambda\; \}$

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

:$\backslash frac\{\backslash lambda\; ^2\}\{(-i\; t+\backslash lambda\; )\; (i\; t+\backslash lambda\; )\}\; =\; \backslash frac\{\backslash lambda\; ^2\}\{t^2+\backslash lambda\; ^2\}.$

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

:$\backslash frac\{1\}\{1+\backslash frac\{t^2\}\{\backslash lambda\; ^2\}\}.$

Sargan distributions are a system of distributions of which the Laplace distribution is a core member. A $p$th order Sargan distribution has densityEveritt, B.S. (2002) The Cambridge Dictionary of Statistics, CUP. {{isbn|0-521-81099-X}}Johnson, N.L., Kotz S., Balakrishnan, N. (1994) Continuous Univariate Distributions, Wiley. {{isbn|0-471-58495-9}}. p. 60

:$f\_p(x)=\backslash tfrac\{1\}\{2\}\; \backslash exp(-\backslash alpha\; |x|)\; \backslash frac\{\backslash displaystyle\; 1+\backslash sum\_\{j=1\}^p\; \backslash beta\_j\; \backslash alpha^j\; |x|^j\}\{\backslash displaystyle\; 1+\backslash sum\_\{j=1\}^p\; j!\backslash beta\_j\},$

for parameters $\backslash alpha\; \backslash ge\; 0,\; \backslash beta\_j\; \backslash ge\; 0$. The Laplace distribution results for $p\; =\; 0$.

Given $n$ independent and identically distributed samples $x\_1,\; x\_2,\; ...,\; x\_n$, the maximum likelihood (MLE) estimator of $\backslash mu$ is the sample median,{{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 = 135–136

| doi = 10.2307/2683252

| jstor = 2683252

| publisher = American Statistical Association

}}

:$\backslash hat\{\backslash mu\}\; =\; \backslash mathrm\{med\}(x).$

The MLE estimator of $b$ is the mean absolute deviation from the median,{{citation needed|date=August 2022}}

:$\backslash hat\{b\}\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i\; =\; 1\}^\{n\}\; |x\_i\; -\; \backslash hat\{\backslash mu\}|.$

revealing a link between the Laplace distribution and least absolute deviations.

A correction for small samples can be applied as follows:

:$\backslash hat\{b\}^*\; =\; \backslash hat\{b\}\; \backslash cdot\; n/(n-2)$

(see: exponential distribution#Parameter estimation).

The Laplacian distribution has been used in speech recognition to model priors on DFT coefficients {{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

}} and in JPEG image compression to model AC coefficients {{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

}} generated by a 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

• In regression analysis, the least absolute deviations estimate arises as the maximum likelihood estimate if the errors have a Laplace distribution.
• The Lasso can be thought of as a Bayesian regression with a Laplacian prior for the coefficients.{{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 }}
• 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 positions 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.{{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 }}{{cite book|title=General Equilibrium Option Pricing Method: Theoretical and Empirical Study|page=70|author=Chen, Jian|year=2018|publisher=Springer|isbn=9789811074288}}

: 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.[https://www.waterlog.info/composite.htm A collection of composite distributions]

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

Given a random variable $U$ drawn from the uniform distribution in the interval $\backslash left(-1/2,\; 1/2\backslash right)$, the random variable

:$X=\backslash mu\; -\; b\backslash ,\backslash sgn(U)\backslash ,\backslash ln(1\; -\; 2|U|)$

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

A $\backslash textrm\{Laplace\}(0,\; b)$ variate can also be generated as the difference of two i.i.d. $\backslash textrm\{Exponential\}(1/b)$ random variables. Equivalently, $\backslash textrm\{Laplace\}(0,1)$ can also be generated as the logarithm of the ratio of two i.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 the central limit theorem.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{{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}}

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.{{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 }}

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

{{Reflist|30em}}

• {{springer|title=Laplace distribution|id=p/l057460}}

{{ProbDistributions|continuous-infinite}}

{{DEFAULTSORT:Laplace Distribution}}

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