Generalized extreme value distribution

Notation	${\textrm {GEV}}(\mu ,\,\sigma ,\,\xi )$ $\textrm{GEV}(\mu,\,\sigma,\,\xi)$
Parameters	μ ∈ R — location, σ > 0 — scale, ξ ∈ R — shape.
Support	x ∈ [ μ − σ / ξ, +∞) when ξ > 0, x ∈ (−∞, +∞) when ξ = 0, x ∈ (−∞, μ − σ / ξ ] when ξ < 0.
PDF	${\frac {1}{\sigma }}\,t(x)^{\xi +1}e^{-t(x)},$ $\frac{1}{\sigma}\,t(x)^{\xi+1}e^{-t(x)},$ where $t(x)={\begin{cases}{\big (}1+({\tfrac {x-\mu }{\sigma }})\xi {\big )}^{-1/\xi }&{\textrm {if}}\ \xi \neq 0\\e^{-(x-\mu )/\sigma }&{\textrm {if}}\ \xi =0\end{cases}}$ $t(x) = \begin{cases}\big(1+(\tfrac{x-\mu}{\sigma})\xi\big)^{-1/\xi} & \textrm{if}\ \xi\neq0 \\ e^{-(x-\mu)/\sigma} & \textrm{if}\ \xi=0\end{cases}$
CDF	$e^{-t(x)},\,$ $e^{-t(x)},\,$ for x ∈ support
Mean	${\begin{cases}\mu +\sigma {\frac {\Gamma (1-\xi )-1}{\xi }}&{\text{if}}\ \xi \neq 0,\xi <1,\\\mu +\sigma \,\gamma &{\text{if}}\ \xi =0,\\\infty &{\text{if}}\ \xi \geq 1,\end{cases}}$ $\begin{cases}\mu + \sigma\frac{\Gamma(1-\xi)-1}{\xi} & \text{if}\ \xi\neq 0,\xi<1,\\ \mu + \sigma\,\gamma & \text{if}\ \xi=0,\\ \infty & \text{if}\ \xi\geq 1,\end{cases}$ where $\gamma$ $\gamma$ is Euler’s constant.
Median	${\begin{cases}\mu +\sigma {\frac {(\ln 2)^{-\xi }-1}{\xi }}&{\text{if}}\ \xi \neq 0,\\\mu -\sigma \ln \ln 2&{\text{if}}\ \xi =0.\end{cases}}$ $\begin{cases}\mu + \sigma \frac{(\ln2)^{-\xi}-1}{\xi} & \text{if}\ \xi\neq0,\\ \mu - \sigma \ln\ln2 & \text{if}\ \xi=0.\end{cases}$
Mode	${\begin{cases}\mu +\sigma {\frac {(1+\xi )^{-\xi }-1}{\xi }}&{\text{if}}\ \xi \neq 0,\\\mu &{\text{if}}\ \xi =0.\end{cases}}$ $\begin{cases}\mu + \sigma \frac{(1+\xi)^{-\xi}-1}{\xi} & \text{if}\ \xi\neq0,\\ \mu & \text{if}\ \xi=0.\end{cases}$
Variance	${\begin{cases}\sigma ^{2}\,(g_{2}-g_{1}^{2})/\xi ^{2}&{\text{if}}\ \xi \neq 0,\xi <{\frac {1}{2}},\\\sigma ^{2}\,{\frac {\pi ^{2}}{6}}&{\text{if}}\ \xi =0,\\\infty &{\text{if}}\ \xi \geq {\frac {1}{2}},\end{cases}}$ $\begin{cases}\sigma^2\,(g_2-g_1^2)/\xi^2 & \text{if}\ \xi\neq0,\xi<\frac12,\\ \sigma^2\,\frac{\pi^2}{6} & \text{if}\ \xi=0, \\ \infty & \text{if}\ \xi\geq\frac12,\end{cases}$ where g_k = Γ(1 − kξ).
Skewness	${\begin{cases}{\frac {g_{3}-3g_{1}g_{2}+2g_{1}^{3}}{(g_{2}-g_{1}^{2})^{3/2}}}&{\text{if}}\ \xi >0,\\-{\frac {g_{3}-3g_{1}g_{2}+2g_{1}^{3}}{(g_{2}-g_{1}^{2})^{3/2}}}&{\text{if}}\ \xi <0,\\{\frac {12{\sqrt {6}}\zeta (3)}{\pi ^{3}}}&{\text{if}}\ \xi =0.\end{cases}}$ $\begin{cases}\frac{g_3-3g_1g_2+2g_1^3}{(g_2-g_1^2)^{3/2}} & \text{if}\ \xi >0,\\ -\frac{g_3-3g_1g_2+2g_1^3}{(g_2-g_1^2)^{3/2}} & \text{if}\ \xi <0,\\ \frac{12 \sqrt{6} \zeta(3)}{\pi^3} & \text{if}\ \xi=0.\end{cases}$ where $\zeta (x)$ $\zeta (x)$ is Riemann zeta function
Ex. kurtosis	${\begin{cases}{\frac {g_{4}-4g_{1}g_{3}+6g_{2}g_{1}^{2}-3g_{1}^{4}}{(g_{2}-g_{1}^{2})^{2}}}-3&{\text{if}}\ \xi \neq 0,\xi <{\frac {1}{4}},\\{\frac {12}{5}}&{\text{if}}\ \xi =0,\\\infty &{\text{if}}\ \xi \geq {\frac {1}{4}}.\end{cases}}$ $\begin{cases}\frac{g_4-4g_1g_3+6g_2g_1^2-3g_1^4}{(g_2-g_1^2)^{2}}-3 & \text{if}\ \xi\neq0,\xi<\frac{1}{4},\\ \frac{12}{5} & \text{if}\ \xi=0,\\ \infty & \text{if}\ \xi\ge\frac{1}{4}.\end{cases}$
Entropy	$\log(\sigma )\,+\,\gamma \xi \,+\,(\gamma +1)$ $\log(\sigma)\,+\,\gamma\xi\,+\,(\gamma+1)$
MGF
CF

${\frac {1}{\sigma }}\,t(x)^{\xi +1}e^{-t(x)},$ $\frac{1}{\sigma}\,t(x)^{\xi+1}e^{-t(x)},$

${\begin{cases}\mu +\sigma {\frac {\Gamma (1-\xi )-1}{\xi }}&{\text{if}}\ \xi \neq 0,\xi <1,\\\mu +\sigma \,\gamma &{\text{if}}\ \xi =0,\\\infty &{\text{if}}\ \xi \geq 1,\end{cases}}$ $\begin{cases}\mu + \sigma\frac{\Gamma(1-\xi)-1}{\xi} & \text{if}\ \xi\neq 0,\xi<1,\\ \mu + \sigma\,\gamma & \text{if}\ \xi=0,\\ \infty & \text{if}\ \xi\geq 1,\end{cases}$

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Wikipedia