Laplace distribution

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	${\displaystyle {\begin{cases}{\frac {1}{2}}\exp \left({\frac {x-\mu }{b}}\right)&{\text{if }}x\leq \mu \\[8pt]1-{\frac {1}{2}}\exp \left(-{\frac {x-\mu }{b}}\right)&{\text{if }}x\geq \mu \end{cases}}}$
Quantile	${\displaystyle {\begin{cases}\mu +b\ln \left(2F\right)&{\text{if }}F\leq {\frac {1}{2}}\\[8pt]\mu -b\ln \left(2-2F\right)&{\text{if }}F\geq {\frac {1}{2}}\end{cases}}}$
Mean	${\displaystyle \mu }$
Median	${\displaystyle \mu }$
Mode	${\displaystyle \mu }$
Variance	${\displaystyle 2b^{2}}$
MAD	${\displaystyle b}$
Skewness	${\displaystyle 0}$
Excess kurtosis	${\displaystyle 3}$
Entropy	${\displaystyle \log(2be)}$
MGF	${\displaystyle {\frac {\exp(\mu t)}{1-b^{2}t^{2}}}{\text{ for }}|t|<1/b}$
CF	${\displaystyle {\frac {\exp(\mu it)}{1+b^{2}t^{2}}}}$
Expected shortfall	${\displaystyle {\begin{cases}\mu +b\left({\frac {p}{1-p}}\right)(1-\ln(2p))&,p<.5\\\mu +b\left(1-\ln \left(2(1-p)\right)\right)&,p\geq .5\end{cases}}}$[1]

In

evaluated over the time scale also have a Laplace distribution.

Definitions

Probability density function

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

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

where ${\displaystyle \mu }$ 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.

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 ${\displaystyle \mu }$, 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.

Cumulative distribution function

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:

${\displaystyle {\begin{aligned}F(x)&=\int _{-\infty }^{x}\!\!f(u)\,\mathrm {d} u={\begin{cases}{\frac {1}{2}}\exp \left({\frac {x-\mu }{b}}\right)&{\mbox{if }}x<\mu \\1-{\frac {1}{2}}\exp \left(-{\frac {x-\mu }{b}}\right)&{\mbox{if }}x\geq \mu \end{cases}}\\&={\tfrac {1}{2}}+{\tfrac {1}{2}}\operatorname {sgn}(x-\mu )\left(1-\exp \left(-{\frac {|x-\mu |}{b}}\right)\right).\end{aligned}}}$

The inverse cumulative distribution function is given by

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

Properties

Moments

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

Related distributions

• If ${\displaystyle X\sim {\textrm {Laplace}}(\mu ,b)}$ then ${\displaystyle kX+c\sim {\textrm {Laplace}}(k\mu +c,|k|b)}$.
• If ${\displaystyle X\sim {\textrm {Laplace}}(0,1)}$ then ${\displaystyle bX\sim {\textrm {Laplace}}(0,b)}$.
• If ${\displaystyle X\sim {\textrm {Laplace}}(0,b)}$ then ${\displaystyle \left|X\right|\sim {\textrm {Exponential}}\left(b^{-1}\right)}$ (exponential distribution).
• If ${\displaystyle X,Y\sim {\textrm {Exponential}}(\lambda )}$ then ${\displaystyle X-Y\sim {\textrm {Laplace}}\left(0,\lambda ^{-1}\right)}$．
• If ${\displaystyle X\sim {\textrm {Laplace}}(\mu ,b)}$ then ${\displaystyle \left|X-\mu \right|\sim {\textrm {Exponential}}(b^{-1})}$.
• If ${\displaystyle X\sim {\textrm {Laplace}}(\mu ,b)}$ then ${\displaystyle X\sim {\textrm {EPD}}(\mu ,b,1)}$ (
  exponential power distribution
  ).
• If ${\displaystyle X_{1},...,X_{4}\sim {\textrm {N}}(0,1)}$ (normal distribution) then ${\displaystyle X_{1}X_{2}-X_{3}X_{4}\sim {\textrm {Laplace}}(0,1)}$ and ${\displaystyle (X_{1}^{2}-X_{2}^{2}+X_{3}^{2}-X_{4}^{2})/2\sim {\textrm {Laplace}}(0,1)}$.
• If ${\displaystyle X_{i}\sim {\textrm {Laplace}}(\mu ,b)}$ then ${\displaystyle {\frac {\displaystyle 2}{b}}\sum _{i=1}^{n}|X_{i}-\mu |\sim \chi ^{2}(2n)}$ (chi-squared distribution).
• If ${\displaystyle X,Y\sim {\textrm {Laplace}}(\mu ,b)}$ then ${\displaystyle {\tfrac {|X-\mu |}{|Y-\mu |}}\sim \operatorname {F} (2,2)}$. (F-distribution)
• If ${\displaystyle X,Y\sim {\textrm {U}}(0,1)}$ (
  uniform distribution
  ) then ${\displaystyle \log(X/Y)\sim {\textrm {Laplace}}(0,1)}$.
• If ${\displaystyle X\sim {\textrm {Exponential}}(\lambda )}$ and ${\displaystyle Y\sim {\textrm {Bernoulli}}(0.5)}$ (Bernoulli distribution) independent of ${\displaystyle X}$, then ${\displaystyle X(2Y-1)\sim {\textrm {Laplace}}\left(0,\lambda ^{-1}\right)}$.
• If ${\displaystyle X\sim {\textrm {Exponential}}(\lambda )}$ and ${\displaystyle Y\sim {\textrm {Exponential}}(\nu )}$ independent of ${\displaystyle X}$, then ${\displaystyle \lambda X-\nu Y\sim {\textrm {Laplace}}(0,1)}$．
• If ${\displaystyle X}$ has a Rademacher distribution and ${\displaystyle Y\sim {\textrm {Exponential}}(\lambda )}$ then ${\displaystyle XY\sim {\textrm {Laplace}}(0,1/\lambda )}$.
• If ${\displaystyle V\sim {\textrm {Exponential}}(1)}$ and ${\displaystyle Z\sim N(0,1)}$ independent of ${\displaystyle V}$, then ${\displaystyle X=\mu +b{\sqrt {2V}}Z\sim \mathrm {Laplace} (\mu ,b)}$.
• If ${\displaystyle X\sim {\textrm {GeometricStable}}(2,0,\lambda ,0)}$ (geometric stable distribution) then ${\displaystyle X\sim {\textrm {Laplace}}(0,\lambda )}$.
• The Laplace distribution is a limiting case of the hyperbolic distribution.
• If ${\displaystyle X|Y\sim {\textrm {N}}(\mu ,Y^{2})}$ with ${\displaystyle Y\sim {\textrm {Rayleigh}}(b)}$ (Rayleigh distribution) then ${\displaystyle X\sim {\textrm {Laplace}}(\mu ,b)}$. Note that if ${\displaystyle Y\sim {\textrm {Rayleigh}}(b)}$, then ${\displaystyle Y^{2}\sim {\textrm {Gamma}}(1,2b^{2})}$ with ${\displaystyle {\textrm {E}}(Y^{2})=2b^{2}}$, which in turn equals the exponential distribution ${\displaystyle {\textrm {Exp}}(1/(2b^{2}))}$.
• Given an integer ${\displaystyle n\geq 1}$, if ${\displaystyle X_{i},Y_{i}\sim \Gamma \left({\frac {1}{n}},b\right)}$ (gamma distribution, using ${\displaystyle k,\theta }$ characterization), then ${\displaystyle \sum _{i=1}^{n}\left({\frac {\mu }{n}}+X_{i}-Y_{i}\right)\sim {\textrm {Laplace}}(\mu ,b)}$ (infinite divisibility)^[2]
• 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.

Probability of a Laplace being greater than another

Let ${\displaystyle X,Y}$ be independent laplace random variables: ${\displaystyle X\sim {\textrm {Laplace}}(\mu _{X},b_{X})}$ and ${\displaystyle Y\sim {\textrm {Laplace}}(\mu _{Y},b_{Y})}$, and we want to compute ${\displaystyle P(X>Y)}$.

The probability of ${\displaystyle P(X>Y)}$ can be reduced (using the properties below) to ${\displaystyle P(\mu +bZ_{1}>Z_{2})}$, where ${\displaystyle Z_{1},Z_{2}\sim {\textrm {Laplace}}(0,1)}$. This probability is equal to

${\displaystyle P(\mu +bZ_{1}>Z_{2})={\begin{cases}{\frac {b^{2}e^{\mu /b}-e^{\mu }}{2(b^{2}-1)}},&{\text{when }}\mu <0\\1-{\frac {b^{2}e^{-\mu /b}-e^{-\mu }}{2(b^{2}-1)}},&{\text{when }}\mu >0\\\end{cases}}}$

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

${\displaystyle P(\mu +Z_{1}>Z_{2})={\begin{cases}e^{\mu }{\frac {(2-\mu )}{4}},&{\text{when }}\mu <0\\1-e^{-\mu }{\frac {(2+\mu )}{4}},&{\text{when }}\mu >0\\\end{cases}}}$

To compute the case for ${\displaystyle \mu >0}$, note that ${\displaystyle P(\mu +Z_{1}>Z_{2})=1-P(\mu +Z_{1}<Z_{2})=1-P(-\mu -Z_{1}>-Z_{2})=1-P(-\mu +Z_{1}>Z_{2})}$

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

Relation to the exponential distribution

A Laplace random variable can be represented as the difference of two

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 ${\displaystyle X,Y\sim {\textrm {Exponential}}(\lambda )}$. The characteristic functions for ${\displaystyle X,-Y}$ are

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

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 ${\displaystyle Z\sim {\textrm {Laplace}}(0,1/\lambda )}$, which is

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

Sargan distributions

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[3]^[4]

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

for parameters ${\displaystyle \alpha \geq 0,\beta _{j}\geq 0}$. The Laplace distribution results for ${\displaystyle p=0}$.

Statistical inference

Given ${\displaystyle n}$ independent and identically distributed samples ${\displaystyle x_{1},x_{2},...,x_{n}}$, the

(MLE) estimator of ${\displaystyle \mu }$ is the sample median,^[5]

${\displaystyle {\hat {\mu }}=\mathrm {med} (x).}$

The MLE estimator of ${\displaystyle b}$ is the

]

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

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

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

(see: exponential distribution#Parameter estimation).

Occurrence and applications

The Laplacian distribution has been used in speech recognition to model priors on DFT coefficients ^[6] and in JPEG image compression to model AC coefficients ^[7] 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.

• In regression analysis, the least absolute deviations estimate arises as the maximum likelihood estimate if the errors have a Laplace distribution.
• The
  prior for the coefficients.^[9]
• In
  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.^[10][11]

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.^[12]

Random variate generation

Given a random variable ${\displaystyle U}$ drawn from the

in the interval ${\displaystyle \left(-1/2,1/2\right)}$, the random variable

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

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

A ${\displaystyle {\textrm {Laplace}}(0,b)}$

${\displaystyle {\textrm {Exponential}}(1/b)}$ random variables. Equivalently, ${\displaystyle {\textrm {Laplace}}(0,1)}$ can also be generated as the 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.^[13][14]

See also

• 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)

References

External links

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