Fréchet distribution

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Fréchet
Probability density function
Cumulative distribution function
Parameters	$\alpha \in (0,\infty]$ shape
Support	$x > 0$
Probability density function (pdf)	$\alpha \; x^{-1-\alpha} \; e^{-x^{-\alpha}}$
Cumulative distribution function (cdf)	$e^{-x^{-\alpha}}$
Mean	$\Gamma\left(1-\frac{1}{\alpha}\right) \text{ if } \alpha>1$
Median	$\left(\frac{1}{\log_e(2)}\right)^{1/\alpha}$
Mode	$\left(\frac{\alpha}{1+\alpha}\right)^{1/\alpha}$
Variance	$\Gamma\left(1-\frac{2}{\alpha}\right)- \left(\Gamma\left(1-\frac{1}{\alpha}\right)\right)^2\text{ if } \alpha>2$
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

The Fréchet distribution is a special case of the generalized extreme value distribution. It has the cumulative distribution function

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

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

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

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

External links

Publications

Fréchet, M., (1927). "Sur la loi de probabilité de l'écart maximum." Ann. Soc. Polon. Math. 6, 93.
Fisher, R.A., Tippett, L.H.C., (1928). "Limiting forms of the frequency distribution of the largest and smallest member of a sample." Proc. Cambridge Philosophical Society 24:180-190.
Gumbel, E.J. (1958). "Statistics of Extremes." Columbia University Press, New York.

This probability-related article is a stub. You can help Wikipedia by expanding it.

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Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin-Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,8)

Beta prime · Burr · chi-square · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scale inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (-8,8)

Cauchy · extreme value · exponential power · Fisher's z · Fisher-Tippett · generalized hyperbolic · hyperbolic secant · Landau · Laplace · Lévy skew alpha-stable · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · Pearson · Tweedie

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