Fréchet distribution

Fréchet
Probability density function
Cumulative distribution function
Parameters	${\displaystyle \ \alpha \in (0,\infty )\ }$ shape. (Optionally, two more parameters) ${\displaystyle \ s\in (0,\infty )\ }$ scale (default: ${\displaystyle \ s=1\ }$) ${\displaystyle \ m\in (-\infty ,\infty )\ }$ location of minimum (default: ${\displaystyle \ m=0\ }$)
Support	${\displaystyle \ x>m\ }$
PDF	${\displaystyle \ {\frac {\ \alpha \ }{s}}\left({\frac {\ x-m\ }{s}}\right)^{-1-\alpha }~e^{-({\frac {x-m}{s}})^{-\alpha }}\ }$
CDF	${\displaystyle \ e^{-({\frac {x-m}{s}})^{-\alpha }}\ }$
Quantile	${\displaystyle \ \left[\ -\ln p\ \right]^{-{\tfrac {1}{\alpha }}}\ s+m\ }$
Mean	${\displaystyle {\begin{cases}\ m\ +\ s\ \Gamma \left(1-{\tfrac {1}{\alpha }}\right)~~&{\text{for }}~\alpha >1\\\ \infty &{\text{otherwise}}\end{cases}}}$
Median	${\displaystyle \ m\ +\ {\frac {s}{\ {\sqrt {\alpha \ }}\ {\log _{e}(2)}\ }}}$
Mode	${\displaystyle \ m\ +\ s\left({\frac {\alpha }{\ 1+\alpha }}\right)^{1/\alpha \ }}$
Variance	${\displaystyle {\begin{cases}\ s^{2}\left[\ \Gamma \left(1-{\tfrac {2}{\alpha }}\right)-\left[\Gamma \left(1-{\tfrac {1}{\alpha }}\right)\right]^{2}\ \right]~~&{\text{for }}~\alpha >2\\\ \infty &{\text{otherwise}}\end{cases}}}$
Skewness	${\displaystyle {\begin{cases}\ {\frac {\ A\ }{\ {\sqrt {B^{3}\ }}\ }}~~&{\text{for }}~\alpha >3\\\ \infty &{\text{otherwise}}\end{cases}}}$ ${\displaystyle {\begin{aligned}{\text{where}}~~A\ \equiv \ &\Gamma \left(1-{\tfrac {3}{\alpha }}\right)\\&-\ 3\ \Gamma \left(1-{\tfrac {2}{\alpha }}\right)\ \Gamma \left(1-{\tfrac {1}{\alpha }}\right)\\&\quad +\ 2{\Bigl [}\ \Gamma \left(1-{\tfrac {1}{\alpha }}\right)\ {\Bigr ]}^{3}\ \end{aligned}}}$ ${\displaystyle ~\quad {\text{and}}~~B\ \equiv \ \Gamma \left(1-{\tfrac {2}{\alpha }}\right)\ -\ {\Bigr [}\ \Gamma \left(1-{\tfrac {1}{\alpha }}\right)\ {\Bigr ]}^{2}~.}$
Excess kurtosis	${\displaystyle {\begin{cases}\ -6\ +\ {\frac {\ C\ }{\;D^{2}}}~~&{\text{for }}~\alpha >4\\\ \infty &{\text{otherwise}}\end{cases}}}$ ${\displaystyle {\begin{aligned}{\text{where}}~~C\ \equiv \ &\Gamma \left(1-{\tfrac {4}{\alpha }}\right)\\&-\ 4\ \Gamma \left(1-{\tfrac {3}{\alpha }}\right)\ \Gamma \left(1-{\tfrac {1}{\alpha }}\right)\\&\qquad +\ 3\ {\Bigl [}\ \Gamma \left(1-{\tfrac {2}{\alpha }}\right)\ {\Bigr ]}^{2}\ \end{aligned}}}$ ${\displaystyle ~\quad {\text{and}}~~D\ \equiv \ \Gamma \left(1-{\tfrac {2}{\alpha }}\right)\ -\ {\Bigl [}\ \Gamma \left(1-{\tfrac {1}{\alpha }}\right)\ {\Bigr ]}^{2}~.}$
Entropy	${\displaystyle \ 1+{\frac {\gamma }{\alpha }}+\gamma _{e}+\ln \left({\frac {s}{\alpha }}\right)\ ,}$ where ${\displaystyle \ \gamma _{e}\ }$ is the Euler–Mascheroni constant.
MGF	Note: Moment ${\displaystyle \ k\ }$ exists if ${\displaystyle \ \alpha >k\ }$
CF

The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function

${\displaystyle \ \Pr(\ X\leq x\ )=e^{-x^{-\alpha }}~{\text{ if }}~x>0~.}$

where α > 0 is a shape parameter. It can be generalised to include a location parameter m (the minimum) and a scale parameter s > 0 with the cumulative distribution function

${\displaystyle \ \Pr(\ X\leq x\ )=\exp \left[\ -\left({\tfrac {\ x-m\ }{s}}\right)^{-\alpha }\ \right]~~{\text{ if }}~x>m~.}$

Named for Maurice Fréchet who wrote a related paper in 1927, further work was done by Fisher and Tippett in 1928 and by Gumbel in 1958.