Laplace distribution


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 spliced together back-to-back, 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. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution.

Definitions

Probability density function

A random variable has a distribution if its probability density function is
Here, is a location parameter and, which is sometimes referred to as the diversity, is a scale parameter. If and, the positive half-line is exactly an exponential distribution scaled by 1/2.
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. Consequently, the Laplace distribution has fatter tails than the normal distribution.

Cumulative distribution function

The Laplace distribution is easy to integrate due to the use of the absolute value function. Its cumulative distribution function is as follows:
The inverse cumulative distribution function is given by

Properties

Moments

where is the generalized exponential integral function.

Related distributions

A Laplace random variable can be represented as the difference of two iid exponential random variables. 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 can be acquired by multiplying the corresponding characteristic functions.
Consider two i.i.d random variables. The characteristic functions for are
respectively. On multiplying these characteristic functions, the result is
This is the same as the characteristic function for, which is

Sargan distributions

Sargan distributions are a system of distributions of which the Laplace distribution is a core member. A th order Sargan distribution has density
for parameters. The Laplace distribution results for.

Statistical Inference

Estimation of parameters

Given independent and identically distributed samples, the maximum likelihood estimator of is the sample median,
and the maximum likelihood estimator of is the Mean Absolute Deviation from the Median
.

Occurrence and applications

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

Generating values from the Laplace distribution

Given a random variable drawn from the uniform distribution in the interval, the random variable
has a Laplace distribution with parameters and. This follows from the inverse cumulative distribution function given above.
A variate can also be generated as the difference of two i.i.d. random variables. Equivalently, can also be generated as the logarithm of the ratio of two i.i.d. uniform random variables.

History

This distribution is often referred to as Laplace's first law of errors. He published it in 1774 when he noted that the frequency of an error could be expressed as an exponential function of its magnitude once its sign was disregarded.
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.