Post History

I managed to find a [reference](https://www.jstor.org/stable/2237384) to page 241 of the textbook "Mathematical method of Statistics" from H. Cramer, where the PDF of the tau distribution should be defined as

$$p(\tau \mathbin{;} \nu) = \frac{\Gamma(\nu / 2)}{\Gamma\bigl((\nu - 1) / 2\bigr) \sqrt{\nu \pi}} \Big(1 - \frac{\tau^2}{\nu}\Big)^{(\nu - 3) / 2},$$

with support in $\bigl[-\sqrt{\nu}, \sqrt{\nu}\bigr]$ and $\nu > 0$.

I also realised that we can use the [change of variables](https://en.wikipedia.org/wiki/Probability_density_function#Function_of_random_variables_and_change_of_variables_in_the_probability_density_function) formula to transform the formulation from the question into this PDF:
Let $g(t) = t \sqrt{\frac{\nu}{t^2 + \nu - 1}}$, then we have
$$\begin{align}
  p(\tau \mathbin{;} \nu) &= p_t\bigl(g^{-1}(\tau) \mathbin{;} \nu - 1\bigr) |g'(\tau)|^{-1} \\\\
  &= p_t\biggl(\pm \tau \sqrt{\frac{\nu - 1}{\nu - \tau^2}} \mathbin{;} \nu - 1\biggr) \nu \sqrt{\frac{\nu - 1}{(\nu - \tau^2)^3}} \\\\
  &= \frac{\Gamma(\nu / 2)}{\Gamma\bigl((\nu - 1) / 2\bigr) \sqrt{(\nu - 1) \pi}} \Big(1 + \frac{\nu - 1}{\nu - \tau^2} \frac{\tau^2}{\nu - 1}\Big)^{-\nu / 2} \nu \sqrt{\frac{\nu - 1}{(\nu - \tau^2)^3}} \\\\
  &= \frac{\Gamma(\nu / 2)}{\Gamma\bigl((\nu - 1) / 2\bigr) \sqrt{\pi}} \Big(\frac{\nu}{\nu - \tau^2}\Big)^{-\nu / 2} \sqrt{\frac{\nu^2}{(\nu - \tau^2)^3}} \\\\
  &= \frac{\Gamma(\nu / 2)}{\Gamma\bigl((\nu - 1) / 2\bigr) \sqrt{\nu \pi}} \Big(\frac{\nu - \tau^2}{\nu}\Big)^{(\nu - 3) / 2}.
\end{align}$$

PS: Mean and variance of this distribution are 0 and 1, respectively.