Gumbel Left-skewed (for minimum order statistic) Distribution¶

\begin{eqnarray*} f\left(x\right) & = & \exp\left(x-e^{x}\right)\\ F\left(x\right) & = & 1-\exp\left(-e^{x}\right)\\ G\left(q\right) & = & \log\left(-\log\left(1-q\right)\right)\end{eqnarray*}

\[M\left(t\right)=\Gamma\left(1+t\right)\]

Note, that \(\mu\) is negative the mean for the right-skewed distribution. Similar for median and mode. All other moments are the same.