Laplace distribution

Laplace
	Probability density function;
	Cumulative distribution function;
parameters:	location (real); scale (real)
support:
pdf:
cdf:	see text
mean:
median:
mode:
variance:
skewness:
kurtosis:
entropy:
mgf:	for
cf:

From Wikipedia, the free encyclopedia

(Redirected from Laplacian distribution)

Jump to: navigation, search

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.

[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:

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

The inverse cumulative distribution function is given by

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

[edit] 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.

[edit] 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).

[edit] Moments

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

[edit] 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 $\lambda_1 X_1-\lambda_2 X_2 \sim \mathrm{Laplace}\left(0,1\right)\,$ ．
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 $b$ .

[edit] See also

Log-Laplace distribution
Lorentzian distribution (the Fourier transform of the Laplace)

[edit] 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.

[0] 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.

[1]

Laplace distribution

From Wikipedia, the free encyclopedia

Contents

[edit] Characterization

[edit] Probability density function

[edit] Cumulative distribution function

[edit] Generating random variables according to the Laplace distribution

[edit] Parameter estimation

[edit] Moments

[edit] Related distributions

[edit] See also

[edit] References

Views

Personal tools

Navigation

Search

Interaction

Toolbox

Languages

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:	$\frac{\exp(\mu\,i\,t)}{1+b^2\,t^2}\,\!$