To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.

Asymmetric Laplace distribution

From Wikipedia, the free encyclopedia

Asymmetric Laplace
Probability density function

Asymmetric Laplace PDF with m = 0 in red. Note that the κ =  2 and 1/2 curves are mirror images. The κ =  1 curve in blue is the symmetric Laplace distribution.
Cumulative distribution function

Asymmetric Laplace CDF with m = 0 in red.

location (real)
scale (real)

asymmetry (real)
PDF (see article)
CDF (see article)


Ex. kurtosis

In probability theory and statistics, the asymmetric Laplace distribution (ALD) is a continuous probability distribution which is a generalization of the Laplace distribution. Just as the Laplace distribution consists of two exponential distributions of equal scale back-to-back about x = m, the asymmetric Laplace consists of two exponential distributions of unequal scale back to back about x = m, adjusted to assure continuity and normalization. The difference of two variates exponentially distributed with different means and rate parameters will be distributed according to the ALD. When the two rate parameters are equal, the difference will be distributed according to the Laplace distribution.


Probability density function

A random variable has an asymmetric Laplace(m, λ, κ) distribution if its probability density function is[1][2]

where s=sgn(x-m), or alternatively:

Here, m is a location parameter, λ > 0 is a scale parameter, and κ is an asymmetry parameter. When κ = 1, (x-m)s κs simplifies to |x-m| and the distribution simplifies to the Laplace distribution.

Cumulative distribution function

The cumulative distribution function is given by:

Characteristic function

The ALD characteristic function is given by:

For m = 0, the ALD is a member of the family of geometric stable distributions with α = 2. It follows that if and are two distinct ALD characteristic functions with m = 0, then

is also an ALD characteristic function with location parameter . The new scale parameter λ obeys

and the new skewness parameter κ obeys:

Moments, mean, variance, skewness

The n-th moment of the ALD about m is given by

From the binomial theorem, the n-th moment about zero (for m not zero) is then:

where is the generalized exponential integral function

The first moment about zero is the mean:

The variance is:

and the skewness is:

Generating asymmetric Laplace variates

Asymmetric Laplace variates (X) may be generated from a random variate U drawn from the uniform distribution in the interval (-κ,1/κ) by:

where s=sgn(U).

They may also be generated as the difference of two exponential distributions. If X1 is drawn from exponential distribution with mean and rate (m1,λ/κ) and X2 is drawn from an exponential distribution with mean and rate (m2,λκ) then X1 - X2 is distributed according to the asymmetric Laplace distribution with parameters (m1-m2, λ, κ)


The differential entropy of the ALD is

The ALD has the maximum entropy of all distributions with a fixed value (1/λ) of where .

Alternative parametrization

An alternative parametrization is made possible by the characteristic function:

where is a location parameter, is a scale parameter, is an asymmetry parameter. This is specified in Section 2.6.1 and Section 3.1 of Lihn (2015). [3] Its probability density function is

where and . It follows that .

The n-th moment about is given by

The mean about zero is:

The variance is:

The skewness is:

The excess kurtosis is:

For small , the skewness is about . Thus represents skewness in an almost direct way.


  1. ^ Kozubowski, Tomasz J.; Podgorski, Krzysztof (2000). "A Multivariate and Asymmetric Generalization of Laplace Distribution". Computational Statistics. 15 (4): 531. doi:10.1007/PL00022717. S2CID 124839639. Retrieved 2015-12-29.
  2. ^ Jammalamadaka, S. Rao; Kozubowski, Tomasz J. (2004). "New Families of Wrapped Distributions for Modeling Skew Circular Data" (PDF). Communications in Statistics – Theory and Methods. 33 (9): 2059–2074. doi:10.1081/STA-200026570. S2CID 17024930. Retrieved 2011-06-13.
  3. ^ Lihn, Stephen H.-T. (2015). "The Special Elliptic Option Pricing Model and Volatility Smile". SSRN: 2707810. Retrieved 2017-09-05.
This page was last edited on 25 December 2020, at 00:29
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.