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 along the abscissa, although the term 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[citation needed].

Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution.

distribution if its probability density function is where

, which is sometimes referred to as the "diversity", is a scale parameter.

, the positive half-line is exactly an exponential distribution scaled by 1/2.

[2] The probability density function 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

, the Laplace density is expressed in terms of the absolute difference from the mean.

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: The inverse cumulative distribution function is given by Let

be independent laplace random variables:

A Laplace random variable can be represented as the difference of two independent and identically distributed (iid) exponential random variables.

[3] One way to show this is by using the characteristic function approach.

For any set of independent continuous random variables, for any linear combination of those variables, its characteristic function (which uniquely determines the distribution) can be acquired by multiplying the corresponding characteristic functions.

On multiplying these characteristic functions (equivalent to the characteristic function of the sum of the random variables

th order Sargan distribution has density[4][5] for parameters

independent and identically distributed samples

, the maximum likelihood (MLE) estimator of

is the sample median,[6] The MLE estimator of

is the mean absolute deviation from the median,[citation needed] revealing a link between the Laplace distribution and least absolute deviations.

A correction for small samples can be applied as follows: (see: exponential distribution#Parameter estimation).

The Laplacian distribution has been used in speech recognition to model priors on DFT coefficients [7] and in JPEG image compression to model AC coefficients [8] generated by a DCT.

drawn from the uniform distribution in the interval

, the random variable has a Laplace distribution with parameters

This follows from the inverse cumulative distribution function given above.

This distribution is often referred to as "Laplace's first law of errors".

He published it in 1774, modeling the frequency of an error as an exponential function of its magnitude once its sign was disregarded.

Laplace would later replace this model with his "second law of errors", based on the normal distribution, after the discovery of the central limit theorem.

[14][15] Keynes published a paper in 1911 based on his earlier thesis wherein he showed that the Laplace distribution minimised the absolute deviation from the median.