From lecture notes on SDE's. I have calculated a stationary distribution of an SDE from the forward kolmogorov equation. I can not, however, identify this distribution and I want to find the mean and variance. The pdf is.
$f(x)=\frac{(x^2+1)^{-\left(\frac{\sigma^2+\lambda}{\sigma^2} \right)}\Gamma\left( \frac{\sigma^2+\lambda}{\sigma^2}\right)}{\sqrt{\pi}\Gamma\left(-\frac{1}{2}+\frac{\sigma^2+\lambda}{\sigma^2} \right)}$
Where $\lambda,\sigma$ are real scalars and $\Gamma$ is the gamma function. It looks like a beta-distribution, but I cannot wrangle it into a form where it yields it's mean and variance?