Show that if N [ , 1 ] ,

Show that if $\beta \mid \lambda \sim \mathcal{N}[\mu ,{\lambda }^{-1}\mathrm{\Sigma }]$, and $\lambda \sim \mathrm{Gamma}[\alpha /2,\alpha /2]$, then the unconditional distribution of $\beta$ is a multivariate $t$-distribution with parameters $(\mu ,\mathrm{\Sigma },\alpha )$.