Laplace distribution | Vose Software

# Laplace distribution

Format: Laplace(m, s)

If X and Y are two identical independent Expon(s) distributions, and if X is shifted m to the right of Y, then (X-Y) is a Laplace(m, s) distribution. The Laplace distribution has an unusual, symmetric shape with a sharp peak and tails that are longer than the tails of a Normal distribution. The figure below plots a Laplace(0,1) against a Normal(0,1) distribution:

## Uses

The Laplace distribution has found a variety of very specific uses, but they nearly all relate to the fact that it has long tails compared to the Normal distribution. It has recently become quite popular in modeling financial variables (Brownian Laplace motion) like stock returns because of the greater tails. The Laplace distribution is very extensively reviewed in the monograph Kotz et al (2001).

The distribution is named after Pierre Simon, Marquis de Laplace - a man so intelligent that his brain was removed and displayed after his death. When m = 0, and s = 1 we have the standard form of the Laplace distribution, which is also occasionally called 'Poisson's first law of error'. The Laplace distribution is also known as the Double-Exponential distribution (though the Gumbel Extreme Value distribution also takes this name), the Two-Tailed Exponential and the Bilateral Exponential distribution.

Skewness = 0, kurtosis = 6.

## ModelRisk functions added to Microsoft Excel for the Laplace distribution

VoseLaplace generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseLaplaceObject constructs a distribution object for this distribution.

VoseLaplaceProb returns the probability density or cumulative distribution function for this distribution.

VoseLaplaceProb10 returns the log10 of the probability density or cumulative distribution function.

VoseLaplaceFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseLaplaceFitObject constructs a distribution object of this distribution fitted to data.

VoseLaplaceFitP returns the parameters of this distribution fitted to data.