Fisher–Tippett distribution

Notation	${\displaystyle {\textrm {GEV}}(\mu ,\,\sigma ,\,\xi )}$
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	${\displaystyle {\frac {1}{\sigma }}\,t(x)^{\xi +1}e^{-t(x)},}$   where ${\displaystyle t(x)={\begin{cases}{\big (}1+\xi ({\tfrac {x-\mu }{\sigma }}){\big )}^{-1/\xi }&{\textrm {if}}\ \xi \neq 0\\e^{-(x-\mu )/\sigma }&{\textrm {if}}\ \xi =0\end{cases}}}$
CDF	${\displaystyle e^{-t(x)},\,}$   for x ∈ support
Mean	${\displaystyle {\begin{cases}\mu +\sigma (g_{1}-1)/\xi &{\text{if}}\ \xi \neq 0,\xi <1,\\\mu +\sigma \,\gamma &{\text{if}}\ \xi =0,\\\infty &{\text{if}}\ \xi \geq 1,\end{cases}}}$ where gk = Γ(1 − kξ), and ${\displaystyle \gamma }$ is Euler’s constant.
Median	${\displaystyle {\begin{cases}\mu +\sigma {\frac {(\ln 2)^{-\xi }-1}{\xi }}&{\text{if}}\ \xi \neq 0,\\\mu -\sigma \ln \ln 2&{\text{if}}\ \xi =0.\end{cases}}}$
Mode	${\displaystyle {\begin{cases}\mu +\sigma {\frac {(1+\xi )^{-\xi }-1}{\xi }}&{\text{if}}\ \xi \neq 0,\\\mu &{\text{if}}\ \xi =0.\end{cases}}}$
Variance	${\displaystyle {\begin{cases}\sigma ^{2}\,(g_{2}-g_{1}^{2})/\xi ^{2}&{\text{if}}\ \xi \neq 0,\xi <{\frac {1}{2}},\\\sigma ^{2}\,{\frac {\pi ^{2}}{6}}&{\text{if}}\ \xi =0,\\\infty &{\text{if}}\ \xi \geq {\frac {1}{2}},\end{cases}}}$.
Skewness	${\displaystyle {\begin{cases}\operatorname {sgn}(\xi ){\frac {g_{3}-3g_{2}g_{1}+2g_{1}^{3}}{(g_{2}-g_{1}^{2})^{3/2}}}&{\text{if}}\ \xi \neq 0,\xi <{\frac {1}{3}},\\{\frac {12{\sqrt {6}}\,\zeta (3)}{\pi ^{3}}}&{\text{if}}\ \xi =0,\\\infty &{\text{if}}\ \xi \geq {\frac {1}{3}}.\end{cases}}}$ where ${\displaystyle \operatorname {sgn}(x)}$ is the sign function and ${\displaystyle \zeta (x)}$ is the Riemann zeta function
Ex. kurtosis	${\displaystyle {\begin{cases}{\frac {g_{4}-4g_{3}g_{1}-3g_{2}^{2}+12g_{2}g_{1}^{2}-6g_{1}^{4}}{(g_{2}-g_{1}^{2})^{2}}}&{\text{if}}\ \xi \neq 0,\xi <{\frac {1}{4}},\\{\frac {12}{5}}&{\text{if}}\ \xi =0,\\\infty &{\text{if}}\ \xi \geq {\frac {1}{4}}.\end{cases}}}$
Entropy	${\displaystyle \log(\sigma )\,+\,\gamma \xi \,+\,\gamma \,+\,1}$
MGF
CF

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

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

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Wikipedia