t as mixture of Normals

(Based on Rasmus Bååth’s post)

A scaled t distribution, with \mu mean, s scale and \nu degrees of freedom, can be simulated

from a mixture of Normals with \mu mean and precisions following a Gamma distribution:

\begin{aligned} y &\sim \mathcal{N}\left(\mu,\sigma\right) \\\\ \sigma^2 &\sim \mathcal{IG}\left(\frac{\nu}{2},s^2\frac{\nu}{2}\right) \end{aligned}

Since I’ve recently pickep up again the crystal-gsl in my spare time, I’ve decided to replicate the previously mentioned post using a Crystal one-liner.

To simulate 10,000 samples from t_2\left(0,3\right) using the mixture, we can then write:

samples = (0..10000).map { |x|
   Normal.sample 0.0, 1.0/Math.sqrt(Gamma.sample 1.0, 9.0)
}

We can see the mixture distribution (histogram) converging nicely to the (t_2(0,3) (red):