I'm a software developer and I have to implement several distributions in my code. I have successfully found information and implemented Student $t$-distribution, and the next one is Log $t$-distribution which literally has very little information online.

I tried integrating $x F(x)$ to calculate the expected mean, but that ended up a fairly long function that I had no idea how to simplify ( was basing on the LogNormal integration to find mean and variance http://mathworld.wolfram.com/LogNormalDistribution.html)

This is the density function of Log Student $t$-distribution:

$$f(x) = \frac{\Gamma((\nu+1)/2)}{x\Gamma(\nu/2)\sigma\sqrt{\nu\pi}}\left( 1+\frac{1}{\nu}\left( \frac{\log x - \mu}{\sigma}\right)^2\right)^{-(\nu+1)/2}$$

Anyone has any tips or perhaps has some more knowledge on the distribution and could shine some light my way on how to calculate the expected mean and variance for this?

  • 5
    $\begingroup$ This distribution does not have a mean and therefore has infinite variance. BTW, understanding (as is conventional) that "$F$" is the cumulative distribution function, you integrate $1-F(x)$ to find the mean (and there is no such thing as "expected mean" except perhaps in a psychological sense). $\endgroup$
    – whuber
    Jan 26, 2018 at 14:06

1 Answer 1


A log-t-distribution does not have a mean. The variance is therefore infinite.

If you take a sample from a log-t-distribution with high degrees of freedom, the mean can be quite stable, and close to the mean of the log-normal distribution. However, with fewer degrees of freedom (and longer tail) and larger sample size(!), you increase the chance to draw extreme values. It is these long-tailed extreme values which blow up the mean and make it unstable.


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