Log-logistic distribution
\(\mbox{LL}(\alpha,\,\gamma)\)
has a density
$$f(x) =
\displaystyle\frac{\alpha\gamma(\alpha\,x)^{\gamma-1}}{\bigl\{1 +
(\alpha\,x)^{\gamma}\bigr\}^{2}},\quad x>0,$$
and a distribution function
$$F(x) =
\displaystyle 1 - \frac{1}{(1 + (\alpha\,x)^\gamma)}, x>0,$$
where \(\alpha\) and \(\gamma\) are positive
parameters (\(\alpha\) is the inverse of the scale parameter and
\(\gamma\) is the shape parameter).
The mean and the variance are given by
$$
\begin{array}{rcll}
\mbox{E}X & \;=\; & \displaystyle \frac{1}{\alpha}\,\frac{\pi}{\gamma\sin\bigl(\frac{\pi}{\gamma}\bigr)}, &\quad \gamma > 1, \\[4ex]
\mbox{var}X & \;=\; & \displaystyle \frac{1}{\alpha^2}\,
\biggl\{\frac{2\pi}{\gamma\sin\bigl(\frac{2\pi}{\gamma}\bigr)}\,-\,
\frac{\pi^2}{\gamma^2\sin^2\bigl(\frac{\pi}{\gamma}\bigr)}\biggr\},
&\quad \gamma > 2, \\[4ex]
\end{array}
$$