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# Laplace distribution

In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also known as the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together back-to-back. The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution.

## Distribution, density, and quantile function

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

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

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

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) $= \int_0^x \!\!f(u)\,\mathrm{d}u$ $= \left\{\begin{matrix} &\frac12 \exp-\frac{\mu-x}{b} & \mbox{if }x < \mu \\[8pt] 1-\!\!\!\!&\frac12 \exp-\frac{x-\mu}{b} & \mbox{if }x \geq \mu \end{matrix}\right.$ $=0.5\,[1 + \sgn(x-\mu)\,(1-\exp(-|x-\mu|/b))]$

The inverse cumulative distribution function is given by

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

## Variates

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

Last updated: 05-31-2005 21:53:46
03-10-2013 05:06:04