Laplace random variables

The exponential distribution really is a gem. Aside from being correct in the vast majority of cases where it is employed (as opposed to the normal, which statistical hacks will apply to just about any problem), it is so perfect mathematically, that many cool properties just seem to pop out of nowhere.

I was generating my random data sets (see step #1 from yesterday) using Studio-R. There are a number of ways to get a sample of pseudo-random Laplace variables. The simplest of course, is to just get a bunch of exponentials with half the variance you're looking for and reverse the sign on half of them.

>laplace = c(-rexp(5000, 1.02), rexp(5000, 1.02))

This gives you a vector of 10,000 random variables distributed as Laplace with a variance of 2.04 (which yields an inter-quartile range of 1.35, same as N(0,1)). But, there's a cooler trick out there. Because of the memoryless property of the exponential distribution, the difference between two independent exponentials winds up being Laplace. This is fairly easy to derive if you think to look for it (I didn't, I just stumbled across it while looking up other things about Laplace).

Again, in R:

>laplace = rexp(10000, 1.02) - rexp(10000, 1.02)

Of course that's a silly way to do it; it requires double the computation. But, it's also kinda cool. In fact, the three distributions I'm working with can all be generated from exponential and uniform. If E1, E2 are standard exponential (mean & variance of 1) and U is uniform(0, 2π), then:

X1 = E1 * sin(U) ~ N(0,1)
X2 = E1 * cos(U) ~ N(0,1)
(and X1 and X2 are independent even though they use the same E1 and U)

L = E1 - E2 ~ Laplace(2)

C = X1 / X2 ~ Cauchy