Laplace distribution

Laplace

Probability density function
Cumulative distribution function
Parameters	$\mu \,$ location (real) $b>0\,$ scale (real)
Support	$x\in (-\infty ;+\infty )\,$
PDF	${\frac {1}{2\,b}}\exp \left(-{\frac {|x-\mu |}{b}}\right)\,$
CDF	see text
Mean	$\mu \,$
Median	$\mu \,$
Mode	$\mu \,$
Variance	$2\,b^{2}$
Skewness	$0\,$
Excess kurtosis	$3\,$
Entropy	$\log(2\,e\,b)$
MGF	${\frac {\exp(\mu \,t)}{1-b^{2}\,t^{2}}}\,\!$ for $|t|<1/b\,$
CF	${\frac {\exp(\mu \,i\,t)}{1+b^{2}\,t^{2}}}\,\!$

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 distributions (with an additional location parameter) spliced together back-to-back, but the term double exponential distribution is also sometimes used to refer to the Gumbel distribution. The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution.

Characterization

Probability density function

A random variable has a Laplace(μ, b) distribution if its probability density function is

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

$={\frac {1}{2b}}\left\{{\begin{matrix}\exp \left(-{\frac {\mu -x}{b}}\right)&{\mbox{if }}x<\mu \\[8pt]\exp \left(-{\frac {x-\mu }{b}}\right)&{\mbox{if }}x\geq \mu \end{matrix}}\right.$

Here, μ is a location parameter and b > 0 is a scale parameter. If μ = 0 and 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 μ, 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.

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:

$F(x)\,$	$=\int _{-\infty }^{x}\!\!f(u)\,\mathrm {d} u$
	$=\left\{{\begin{matrix}&{\frac {1}{2}}\exp \left({\frac {x-\mu }{b}}\right)&{\mbox{if }}x<\mu \\[8pt]1-\!\!\!\!&{\frac {1}{2}}\exp \left(-{\frac {x-\mu }{b}}\right)&{\mbox{if }}x\geq \mu \end{matrix}}\right.$
	$=0.5\,[1+\operatorname {sgn}(x-\mu )\,(1-\exp(-|x-\mu |/b))].$

The inverse cumulative distribution function is given by

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

Generating random variables according to the Laplace distribution

Given a random variable U drawn from the uniform distribution in the interval (-1/2, 1/2], the random variable

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

has a Laplace distribution with parameters μ and b. This follows from the inverse cumulative distribution function given above.

A Laplace(0, b) variate can also be generated as the difference of two i.i.d. Exponential(1/b) random variables. Equivalently, a Laplace(0, 1) random variable can be generated as the logarithm of the ratio of two iid uniform random variables.

Parameter estimation

Given N independent and identically distributed samples x₁, x₂, ..., x_N, an estimator ${\hat {\mu }}$ of $\mu$ is the sample median, and the maximum likelihood estimator of b is

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

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

Moments

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

Related distributions

• If $X\sim \mathrm {Laplace} (\mu ,b)\,$ then $kX+c\sim \mathrm {Laplace} (k\mu +c,kb)\,$
• If $X\sim \mathrm {Laplace} (0,b)\,$ then $|X|\sim \mathrm {Exponential} ({\tfrac {1}{b}})\,$ (exponential distribution)
• If $X\sim \mathrm {Exponential} ({\tfrac {1}{\lambda }})\,$ and $Y\sim \mathrm {Exponential} ({\tfrac {1}{\lambda }})\,$ then $X-Y\sim \mathrm {Laplace} \left(0,\lambda \right)\,$ ．
• If $X\sim \mathrm {Laplace} (\mu ,{\tfrac {1}{b}})\,$ then $|X-\mu |\sim \mathrm {Exponential} (b)\,$
• If $X\sim \mathrm {Laplace} (\mu ,b)\,$ then $X\sim \mathrm {EPD} (\mu ,\alpha =b,\beta =0)\,$ (Exponential power distribution)
• If $X_{i}\sim \mathrm {Normal} (0,1)\,$ (Normal distribution) for $i={1,2,3,4}\,$ then $X_{1}X_{2}-X_{3}X_{4}\sim \mathrm {Laplace} (0,1)\,$
• If $X_{i}\sim \mathrm {Laplace} (\mu ,b)\,$ then ${\frac {2}{b\sum _{i=1}^{n}|X_{i}-\mu |}}\sim \chi ^{2}(2n)\,$ (Chi-square distribution)
• If $X\sim \operatorname {Laplace} (\mu ,b)$ and $Y\sim \operatorname {Laplace} (\mu ,b)$ then ${\tfrac {|X-\mu |}{|Y-\mu |}}\sim \operatorname {F} (2,2)$ (F-distribution)
• If $X\sim \mathrm {U} (0,1)\,$ and $Y\sim \mathrm {U} (0,1)\,$ (Uniform distribution (continuous)) then $\log {\tfrac {X}{Y}}\sim \mathrm {Laplace} (0,1)\,$
• If $X\sim \mathrm {Exponential} (\lambda )\,$ and ${Y\sim \,}$ $\mathrm {Bernoulli} (0.5)\,$ independent of $X\,$, then $X(2Y-1)\sim \mathrm {Laplace} (0,\lambda ^{-1})\,$ .
• If $X_{1}\sim \mathrm {Exponential} (\lambda _{1})\,$ and $X_{2}\sim \mathrm {Exponential} (\lambda _{2})\,$ independent of $X_{1}\,$, then $\lambda _{1}X_{1}-\lambda _{2}X_{2}\sim \mathrm {Laplace} \left(0,1\right)\,$ ．
• If $V\sim \mathrm {Exponential} (1)\,$ and $Z\sim \mathrm {N} (0,1)\,$ independent of $V$, then $X=\mu +b{\sqrt {2V}}Z\sim \mathrm {Laplace} (\mu ,b)$.
• If $X\sim \mathrm {GeometricStable} (2,0,\lambda ,0)\,$ is a geometric stable distribution with $\alpha$=2, $\beta$=0 and $\mu$=0 then $X\sim \mathrm {Laplace} (0,\lambda )\,$ is a Laplace distribution with $\mu =0$ and b=$\lambda$
• Laplace distribution is the limiting case of Hyperbolic distribution
• If $X|Y\sim \mathrm {Normal} (\mu ,\sigma =Y)\,$ with $Y\sim \mathrm {Rayleigh} (b)\,$ (Rayleigh distribution) then $X\sim \mathrm {Laplace} (\mu ,b)\,$

Relation to the exponential distribution

A Laplace random variable can be represented as the difference of two 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 correspond characteristic functions.

Consider two i.i.d random variables $X_{1},X_{2}\sim \mathrm {Exponential} (\lambda )\;$. The characteristic functions for $X_{1},-X_{2}$ are ${\frac {\lambda }{-it+\lambda }},{\frac {\lambda }{it+\lambda }}$, respectively. On multiplying these characteristic functions (equivalent to the characteristic function of the sum of therandom variables $X_{1}+(-X_{2})$), the result is ${\frac {\lambda ^{2}}{(-it+\lambda )(it+\lambda )}}={\frac {\lambda ^{2}}{t^{2}+\lambda ^{2}}}$.

This is the same as the characteristic function for $Y\sim \mathrm {Laplace} (0,1/\lambda )\;$, which is ${\frac {1}{1+{\frac {t^{2}}{\lambda ^{2}}}}}$.

Sargan distributions

Sargan distributions are a system of distributions of which the Laplace distribution is a core member. A p'th order Sargan distribution has density

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

for parameters α > 0, β_j   ≥ 0. The Laplace distribution results for p=0.

See also

• Log-Laplace distribution
• Cauchy distribution, also called the "Lorentzian distribution" (the Fourier transform of the Laplace)
• Characteristic function (probability theory)