# Laplace

Laplace(m,s) = m+Exponential(SQRT(2)/s)-Exponential(SQRT(2)/s)

Laplace equations

If X and Y are two identical independent Exponential(1/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 a strange shape with a sharp peak and tails that are longer than tails of a Normal distribution. The figure below plots a Laplace(0,1) against a Normal(0,1) distribution: #### Uses

The Laplace has found a variety of very specific uses, but they nearly all relate to the fact that it has long tails.

#### Generation

Crystal Ball does not have this distribution, but it can easily be generated as the difference between two identical Exponential distributions as follows:

The links to the Laplace software specific models are provided here: Laplace Laplace

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.

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