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.