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

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):