# Talk:Logistic distribution/Generalized log-logistic distribution

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