Laplace distribution

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“Double exponential distribution” redirects here. For the similarly-named function, see double exponential function.

Laplace
Probability density function
Cumulative distribution function
Parameters	$\mu\,$ location (real) $b > 0\,$ scale (real)
Support	$x \in (-\infty; +\infty)\,$
Probability density function (pdf)	$\frac{1}{2\,b} \exp \left(-\frac{\|x-\mu\|}b \right) \,$
Cumulative distribution function (cdf)	see text
Mean	$\mu\,$
Median	$\mu\,$
Mode	$\mu\,$
Variance	$2\,b^2$
Skewness	$0\,$
Excess kurtosis	$3\,$
Entropy	$1 + \ln(2\,b)$
Moment-generating function (mgf)	$\frac{\exp(\mu\,t)}{1-b^2\,t^2}\,\!$ for $\|t\|<1/b\,$
Characteristic function	$\frac{\exp(\mu\,i\,t)}{1+b^2\,t^2}\,\!$

In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also 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, as is a Brownian motion evaluated at an exponentially distributed random time.

1 Characterization
- 1.1 Probability density function
- 1.2 Cumulative distribution function
2 Generating Laplace variates
3 Parameter estimation
4 Related distributions
5 References

[edit] Characterization

[edit] 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.

[edit] 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:

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

The inverse cumulative distribution function is given by

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

[edit] Generating Laplace variates

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

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

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) variates. Equivalently, a Laplace(0, 1) variate can be generated as the logarithm of the ratio of two iid uniform variates.

[edit] Parameter estimation

Given N iid samples x₁, x₂, ..., x_N, estimator of $μ$ , $\hat{\mu}$ is the sample median^[1], and the estimator of b is

$\hat{b} = \frac{1}{N} \sum_{i = 1}^{N} |x_i - \hat{\mu}|,$

using the maximum likelihood estimator.

[edit] Related distributions

If $X \sim \mathrm{Laplace}(0,b)\,$ then $|X| \sim \mathrm{Exponential}(b^{-1})\,$ is an exponential distribution.
If $X_1 \sim \mathrm{Exponential}(\lambda_1)\,$ and $X_2 \sim \mathrm{Exponential}(\lambda_2)\,$ then $X_1-X_2 \sim \mathrm{Laplace}\left(0,\frac{\lambda_1+\lambda_2}{2\lambda_1\lambda_2}\right)\,$ .

[edit] References

^ [1] Robert M. Norton,Double Exponential Distribution: Using Calculus to Find a Maximum Likelihood Estimator The American Statistician, Vol. 38, No. 2. (May, 1984), pp. 135-136.

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