Mathematics

The tau-distribution is typically defined in terms of the Student's t-distribution as follows: $$\tau_\nu \sim \sqrt{\frac{\nu \, t_{\nu - 1}^2}{\nu - 1 + t_{\nu - 1}^2}}.$$

I would be interested to compute the mean and variance of this random variable. Because it seems daunting to directly compute expectations from this formula, I started looking for a Probability Density Function (PDF) for this distribution.

Unfortunately, I was not very successful because everyone seems to use the formulation above. That made me wonder whether this formula (implicitly) defines the PDF and I am too stupid to see it, or if there simply is no proper density function. Hence my question: what is the PDF for the tau distribution?

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What is the probability density function for the tau distribution?

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