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Encyclopedia > Laplace distribution
Laplace
Probability density function
Probability density plots of Laplace distributions
Cumulative distribution function
Cumulative distribution plots of Laplace distributions
Parameters location (real)
scale (real)
Support
pdf
cdf see text
Mean
Median
Mode
Variance
Skewness
Kurtosis
Entropy
mgf for
Char. func.

In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also known as the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together back-to-back. 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. Download high resolution version (1300x975, 135 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1300x975, 151 KB) Wikipedia does not have an article with this exact name. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... In probability (and especially gambling), the expected value (or (mathematical) expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds... In probability theory and statistics, the median is a number that separates the highest half of a sample, a population, or a probability distribution from the lowest half. ... In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability. ... In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith... Some mathematicians use the phrase characteristic function synonymously with indicator function. The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A... Probability theory is the mathematical study of probability. ... Statistics is a type of data analysis whose practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... Pierre-Simon Laplace Pierre-Simon, Marquis de Laplace (March 23, 1749 – March 5, 1827) was a French mathematician and astronomer, the discoverer of the Laplace transform and Laplaces equation, which appear in all branches of mathematical physics. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ...


Distribution, density, and quantile function

A random variable has a Laplace(μ, b) distribution if its probability density function is A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

Here, μ is a location parameter and b > 0 is a scale parameter. If μ = 0, the positive half-line is exactly an exponential distribution scaled by 1/2. In statistics, if a family of probabiblity densities parametrized by a scalar- or vector-valued parameter μ is of the form fμ(x) = f(x − μ) then μ is called a location parameter, since its value determines the location of the probability distribution. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ...


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


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: In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... The graph of the absolute value function In mathematics, the absolute value (or modulus) of a real number is its numerical value without regard to its sign. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or...

The inverse cumulative distribution function is given by

Generating Laplace variates

Given a random variate U drawn from the uniform distribution in the interval (-1/2, 1/2], the variate In mathematics, the uniform distributions are simple probability distributions. ...

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. A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ... In mathematics, a logarithm of x with base b may be defined as the following: for the equation bn = x, the logarithm is a function which gives n. ...


Related distributions


  Results from FactBites:
 
Laplace distribution - Wikipedia, the free encyclopedia (394 words)
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.
The pdf 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 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.
  More results at FactBites »

 
 

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