Normal scale mixture representation of Student t ($T=Z/\sqrt{\Lambda /\nu },\Lambda \sim {\chi }_{\nu }^2$)

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.