# What processes could generate Laplace-distributed (double exponential) data or parameters?

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.

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• If $X_1$ and $X_2$ are IID exponential distributions, $X_1 - X_2$ has a Laplace distribution.
• If $X_1, X_2, X_3, X_4$ are IID standard normal distributions, $X_1X_4 - X_2X_3$ has a standard Laplace distribution. So, the determinant of a random $2\times 2$ matrix with IID standard normal entries $\begin{pmatrix}X_1 & X_2 \\\ X_3 & X_4 \end{pmatrix}$ has a Laplace distribution.
• If $X_1, X_2$ are IID uniform on $[0,1]$, then $\log \frac{X_1}{X_2}$ has a standard Laplace distribution.
+1 It may be worth noticing that all three examples are really the same: #2 can be rewritten as $((X_1+X_4)^2 + (X_2+X_3)^2 - [(X_1-X_4)^2 + (X_2-X_3)^2])/4$, a scaled difference of two scaled $\chi^2(2)$ (Exponential) distributions, and #3 is the difference of two Exponential distributions because the $\log(X_i)$ are Exponential. –  whuber Sep 24 '12 at 20:59
@whuber: Thanks for that explanation for why the determinant was the same as the others! I was surprised to see it, since I would have guessed that the density of the determinant would vary smoothly, as it does everywhere except $0$. –  Douglas Zare Sep 24 '12 at 21:03