= Normal scale mixture representation of Student t
{title2=$T=Z/\sqrt{\Lambda/\nu},\quad\Lambda\sim\chi_\nu^2$}
For independent $Z\sim N(0,1)$ and $\Lambda\sim\chi_\nu^2$, the displayed ratio has the <Student t-distribution> with $\nu$ degrees of freedom. Equivalently, $T\mid\Lambda\sim N(0,\nu/\Lambda)$. Integrating the normal conditional density against the <chi-squared distribution> gives a density proportional to $(1+t^2/\nu)^{-(\nu+1)/2}$. A latent scale therefore supplies heavy tails while retaining convenient normal conditional calculations. For $\nu>2$, its variance is $\nu/(\nu-2)$, so its scale parameter is not its standard deviation.
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