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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:	$0\,$
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.

1 Characterization
- 1.1 Probability density function
- 1.2 Cumulative distribution function
2 Generating random variables according to the Laplace distribution
3 Parameter estimation
4 Moments
5 Related distributions
6 See also
7 References

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 pdf 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:

$F(x)\,$	$= \int_{-\infty}^x \!\!f(u)\,\mathrm{d}u$
	$= \left\{\begin{matrix} &\frac12 \exp \left( \frac{x-\mu}{b} \right) & \mbox{if }x < \mu \\[8pt] 1-\!\!\!\!&\frac12 \exp \left( -\frac{x-\mu}{b} \right) & \mbox{if }x \geq \mu \end{matrix}\right.$
	$=0.5\,[1 + \sgn(x-\mu)\,(1-\exp(-\|x-\mu\|/b))].$

The inverse cumulative distribution function is given by

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

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

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

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 $μ$ is the sample median,^[1] 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 $X \sim \mathrm{Laplace}(0,b)\,$ then $|X| \sim \mathrm{Exponential}(b^{-1})\,$ is an exponential distribution.
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 $X = \mu + b \sqrt{2 V}Z \sim \mathrm{Laplace}(\mu,b)$ .
The generalized Gaussian distribution (version 1) equals the Laplace distribution when its shape parameter $β$ is set to 1. The scale parameter $α$ is then equal to $b$ .

References

^ Robert M. Norton (May 1984). "The Double Exponential Distribution: Using Calculus to Find a Maximum Likelihood Estimator". The American Statistician 38 (2): 135–136. doi:10.2307/2683252. http://www.jstor.org/pss/2683252.

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway–Maxwell–Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss–Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule–Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin–Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi–Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell–Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional:Dirac comb · Circular Uniform · bivariate von Mises · Kent · univariate von Mises · von Mises–Fisher · Wrapped normal · Wrapped Cauchy · Wrapped Lévy Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · stable · Tweedie

Some common univariate probability distributions

Continuous	beta • Cauchy • chi-square • exponential • F • gamma • Laplace • log-normal • normal • Pareto • Student's t • uniform • Weibull

Discrete	Bernoulli • binomial • discrete uniform • geometric • hypergeometric • negative binomial • Poisson

List of probability distributions