Laplace distribution
Encyclopedia
In probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the Laplace distribution is a continuous probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 named after Pierre-Simon Laplace
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distribution
Exponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

s (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
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

 evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process
Variance gamma process
In the theory of stochastic processes, a part of the mathematical theory of probability, the variance gamma process , also known as Laplace motion, is a Lévy process determined by a random time change. The process has finite moments distinguishing it from many Lévy processes. There is no diffusion...

 evaluated over the time scale also have a Laplace distribution.

Probability density function

A random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

 has a Laplace(μ, b) distribution if its probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 is


Here, μ is a location parameter
Location parameter
In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution...

 and b > 0 is a scale parameter
Scale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...

. If μ = 0 and b = 1, the positive half-line is exactly an exponential distribution
Exponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

 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 μ, the Laplace density is expressed in terms of the absolute difference
Absolute difference
The absolute difference of two real numbers x, y is given by |x − y|, the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y...

 from the mean. Consequently the Laplace distribution has fatter tails than the normal distribution.

Cumulative distribution function

The Laplace distribution is easy to integrate
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 (if one distinguishes two symmetric cases) due to the use of the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

 function. Its cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

 is as follows: |
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The inverse cumulative distribution function is given by

Generating random variables according to the Laplace distribution

Given a random variable U drawn from the uniform distribution
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...

 in the interval (-1/2, 1/2], the random variable


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
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

 of the ratio of two iid uniform random variables.

Parameter estimation

Given N independent and identically distributed samples x1, x2, ..., xN, an estimator of is the sample median
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...

,
and the maximum likelihood
Maximum likelihood
In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....

 estimator of b is
(revealing a link between the Laplace distribution and least absolute deviations
Least absolute deviations
Least absolute deviations , also known as Least Absolute Errors , Least Absolute Value , or the L1 norm problem, is a mathematical optimization technique similar to the popular least squares technique that attempts to find a function which closely approximates a set of data...

).

Moments


Related distributions

  • If then
  • If then (exponential distribution
    Exponential distribution
    In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

    )
  • If and then
  • If then
  • If then (Exponential power distribution)
  • If (Normal distribution) for then
  • If then (Chi-squared distribution)
  • If and then (F-distribution)
  • If and (Uniform distribution (continuous)
    Uniform distribution (continuous)
    In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...

    ) then
  • If and  independent of , then .
  • If and independent of , then
  • If and independent of , then .
  • If is a geometric stable distribution with =2, =0 and =0 then is a Laplace distribution with and b=
  • Laplace distribution is the limiting case of Hyperbolic distribution
    Hyperbolic distribution
    The hyperbolic distribution is a continuous probability distribution that is characterized by the fact that the logarithm of the probability density function is a hyperbola. Thus the distribution decreases exponentially, which is more slowly than the normal distribution...

  • If with (Rayleigh distribution) then

Relation to the exponential distribution

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
Characteristic function
In mathematics, characteristic function can refer to any of several distinct concepts:* The most common and universal usage is as a synonym for indicator function, that is the function* In probability theory, the characteristic function of any probability distribution on the real line is given by...

 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 correspond characteristic functions.

Consider two i.i.d random variables . The characteristic functions for are , respectively. On multiplying these characteristic functions (equivalent to the characteristic function of the sum of therandom variables ), 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 pth order Sargan distribution has density
for parameters α > 0, βj   ≥ 0. The Laplace distribution results for p=0.

Applications

The Laplacian distribution has been used in speech recognition to model priors on DFT
DFT
DFT may stand for:*Discrete Fourier transform*Decision field theory*Density functional theory*Demand Flow Technology*The United Kingdom's Department for Transport*Design For Test*Deareating Feed Tank*Digital Film Technology maker of the Spirit DataCine...

 coefficients.

See also

  • Log-Laplace distribution
    Log-Laplace distribution
    In probability theory and statistics, the log-Laplace distribution is the probability distribution of a random variable whose logarithm has a Laplace distribution. If X has a Laplace distribution with parameters μ and b, then Y = eX has a log-Laplace distribution...

  • Cauchy distribution
    Cauchy distribution
    The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...

    , also called the "Lorentzian distribution" (the Fourier transform of the Laplace)
  • Characteristic function (probability theory)
    Characteristic function (probability theory)
    In probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...

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