# Talk:Logistic distribution/Generalized log-logistic distribution

The Generalized log-logistic distribution (GLL) has three parameters $\mu,\sigma \,$ and $\xi$.
Parameters $\mu \in (-\infty,\infty) \,$ location (real) $\sigma \in (0,\infty) \,$ scale (real) $\xi\in (-\infty,\infty) \,$ shape (real) $x \geqslant \mu -\sigma/\xi\,\;(\xi \geqslant 0)$ $x \leqslant \mu-\sigma/\xi\,\;(\xi < 0)$ $\frac{(1+\xi z)^{-(1/\xi +1)}}{\sigma\left(1 + (1+\xi z)^{-1/\xi}\right)^2}$ where $z=(x-\mu)/\sigma\,$ $\left(1+(1 + \xi z)^{-1/\xi}\right)^{-1} \,$ where $z=(x-\mu)/\sigma\,$ $\mu + \frac{\sigma}{\xi}(\alpha \csc(\alpha)-1)$ where $\alpha= \pi \xi\,$ $\mu \,$ $\mu + \frac{\sigma}{\xi}\left[\left(\frac{1-\xi}{1+\xi}\right)^\xi - 1 \right]$ $\frac{\sigma^2}{\xi^2}[2\alpha \csc(2 \alpha) - (\alpha \csc(\alpha))^2]$ where $\alpha= \pi \xi\,$
$F_{(\xi,\mu,\sigma)}(x) = \left(1 + \left(1+ \frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}\right)^{-1}$
for $1 + \xi(x-\mu)/\sigma \geqslant 0$, where $\mu\in\mathbb R$ is the location parameter, $\sigma>0 \,$ the scale parameter and $\xi\in\mathbb R$ the shape parameter. Note that some references give the "shape parameter" as $\kappa = - \xi \,$.
$\frac{\left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-(1/\xi +1)}} {\sigma\left[1 + \left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}\right]^2} .$
again, for $1 + \xi(x-\mu)/\sigma \geqslant 0.$