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| ).