Laplace distribution

Probability distribution
name =Laplace
type =density
pdf_

cdf_

parameters =$mu,$ location (real)
$b\; >\; 0,$ scale (real)
support =$x\; in\; (-infty;\; +infty),$
pdf =$frac\{1\}\{2,b\}\; exp\; left(-fracThe\; inverse\; cumulative\; distribution\; function\; is\; given\; by$

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

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.

Parameter estimation

Given "N" independent and identically distributed samples "x₁, x₂, ..., x_N", an estimator $hat\{mu\}$ of $mu$ is the sample median, [Cite journal
author = Robert M. Norton
title = The Double Exponential Distribution: Using Calculus to Find a Maximum Likelihood Estimator
journal = The American Statistician
volume = 38
issue = 2
month = May
year = 1984
pages = 135–136
url = http://www.jstor.org/pss/2683252
doi = 10.2307/2683252] and the estimator of "b" is:$hat\{b\}\; =\; frac\{1\}\{N\}\; sum\_\{i\; =\; 1\}^\{N\}\; |x\_i\; -\; hat\{mu\}|,$using the maximum likelihood estimator.

Moments

:igg) sum_{k=0}^r igg [{frac{r!}{k! (r-k)! b^k mu^{(r-k)} k! {1 + (-1)^k}igg]

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 $lambda\_1\; X\_1-lambda\_2\; X\_2\; sim\; mathrm\{Laplace\}left(0,1\; ight),$ ．

ee also

Log-Laplace distribution

References