Fréchet distribution

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

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

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	${\displaystyle e^{-({\frac {x-m}{s}})^{-\alpha }}}$
Mean	${\displaystyle {\begin{cases}\ m+s\Gamma \left(1-{\frac {1}{\alpha }}\right)&{\text{for }}\alpha >1\\\ \infty &{\text{otherwise}}\end{cases}}}$
Median	${\displaystyle m+{\frac {s}{\sqrt[{\alpha }]{\log _{e}(2)}}}}$
Mode	${\displaystyle m+s\left({\frac {\alpha }{1+\alpha }}\right)^{1/\alpha }}$
Variance	${\displaystyle {\begin{cases}\ s^{2}\left(\Gamma \left(1-{\frac {2}{\alpha }}\right)-\left(\Gamma \left(1-{\frac {1}{\alpha }}\right)\right)^{2}\right)&{\text{for }}\alpha >2\\\ \infty &{\text{otherwise}}\end{cases}}}$
Skewness	${\displaystyle {\begin{cases}\ {\frac {\Gamma \left(1-{\frac {3}{\alpha }}\right)-3\Gamma \left(1-{\frac {2}{\alpha }}\right)\Gamma \left(1-{\frac {1}{\alpha }}\right)+2\Gamma ^{3}\left(1-{\frac {1}{\alpha }}\right)}{\sqrt {\left(\Gamma \left(1-{\frac {2}{\alpha }}\right)-\Gamma ^{2}\left(1-{\frac {1}{\alpha }}\right)\right)^{3}}}}&{\text{for }}\alpha >3\\\ \infty &{\text{otherwise}}\end{cases}}}$
Ex. kurtosis	${\displaystyle {\begin{cases}\ -6+{\frac {\Gamma \left(1-{\frac {4}{\alpha }}\right)-4\Gamma \left(1-{\frac {3}{\alpha }}\right)\Gamma \left(1-{\frac {1}{\alpha }}\right)+3\Gamma ^{2}\left(1-{\frac {2}{\alpha }}\right)}{\left[\Gamma \left(1-{\frac {2}{\alpha }}\right)-\Gamma ^{2}\left(1-{\frac {1}{\alpha }}\right)\right]^{2}}}&{\text{for }}\alpha >4\\\ \infty &{\text{otherwise}}\end{cases}}}$
Entropy	${\displaystyle 1+{\frac {\gamma }{\alpha }}+\gamma +\ln \left({\frac {s}{\alpha }}\right)}$, where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant.
MGF	Note: Moment ${\displaystyle k}$ exists if ${\displaystyle \alpha >k}$
CF

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

${\displaystyle \Pr(X\leq x)=e^{-\left({\frac {x-m}{s}}\right)^{-\alpha }}{\text{ if }}x>m.}$

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.