Lots of distributions have “origin myths”, or examples of physical processes that they describe well:
- You can get normally distributed data from sums of uncorrelated errors via the Central Limit Theorem
- You can get binomially distributed data from independent coin flips, or Poisson-distributed variables from a limit of that process
- You can get exponentially distributed data from waiting times under a constant decay rate.
And so on.
But what about the Laplace distribution? It’s useful for L1 regularization and LAD regression, but it’s hard for me to think of a situation where one should actually expect to see it in nature. Diffusion would be Gaussian, and all the examples I can think of with exponential distributions (e.g. waiting times) involve non-negative values.
At the bottom of the Wikipedia page you linked are a few examples:
If X1 and X2 are IID exponential distributions, X1−X2 has a Laplace distribution.
If X1,X2,X3,X4 are IID standard normal distributions, X1X4−X2X3 has a standard Laplace distribution. So, the determinant of a random 2×2 matrix with IID standard normal entries (X1X2 X3X4) has a Laplace distribution.
If X1,X2 are IID uniform on [0,1], then logX1X2 has a standard Laplace distribution.