Fréchet distribution: Definition from Answers.com

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 α>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

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.

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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 the whole real line (-∞,∞)

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · 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 · mixture · Pearson · stable · Tweedie

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		Statistics Dictionary. A Dictionary of Statistics. Second edition revised. Copyright © Oxford University Press, 2008. All rights reserved. Read more
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