While I haven't worked this out for Gaussian random variables, it is easy to get the distribution of the product of n independent random variables uniform on [0,1]. It is p[n](x) = (-log x)^(n-1) / (n-1)! The singularity at 0 should be the same for a Gaussian, and indeed for x near 0 (and > 0) K[0](x) ~ -log x + constant. -- Gene From: Dan Asimov <asimov@msri.org> To: Eugene Salamin <gene_salamin@yahoo.com>; math-fun <math-fun@mailman.xmission.com> Sent: Saturday, March 5, 2016 12:53 PM Subject: Re: [math-fun] How to generate a random variable with a BesselK[0] distribution Interesting (though I've never needed a Bessel distribution). I wonder what the distribution is of the product of n i.i.d. N(0,1) random variables is. —Dan
On Mar 5, 2016, at 12:25 PM, Eugene Salamin via math-fun <math-fun@mailman.xmission.com> wrote:
Suppose, for some reason, you need to generate random variables whose normalized probability distribution is
P(z) = (1/pi) BesselK[0] ( |z| ).
Here's a clever way to do it without having to muck with Bessel functions. Let x and y be independent Gaussian random variables with zero mean and unit variance. Then z = xy has the desired distribution.
Furthermore, the sum of n such products, has distribution
P[n](z) = (1 / (sqrt(pi) Gamma(n/2)) (|z| / 2)^((n-1)/2) BesselK[((n-1)/2] ( |z| ).