Laplace(m,s) = m+Exponential(SQRT(2)/s)-Exponential(SQRT(2)/s)
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:
| Laplace(\mu ,\sigma)=\mu + Exponential\Big(\sqrt{2}/\sigma \Big)- Exponential\Big(\sqrt{2}/\sigma \Big) |
You can find an example in Laplace model.
The links to the Laplace software specific models are provided here:
Comments
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.
