Consider again Example 2.

8, where we have a normal model with improper prior \(g(\boldsymbol{\theta})\) \(=g\left(\mu, \sigma^{2}\right) \propto 1 / \sigma^{2}\). Show that the prior predictive pdf is an improper density \(g(x) \propto 1\), but that the posterior predictive density is

\[ g(x \mid \tau) \propto\left(1+\frac{\left(x-\bar{x}_{n}\right)^{2}}{(n+1) S_{n}^{2}}\right)^{-n / 2} \]

Deduce that \(\frac{X-\bar{x}_{n}}{S_{n} \sqrt{(n+1) /(n-1)}} \sim t_{n-1}\).