Type-2 Gumbel distribution

You don't need to be Editor-In-Chief to add or edit content to WikiDoc. You can begin to add to or edit text on this WikiDoc page by clicking on the edit button at the top of this page. Next enter or edit the information that you would like to appear here. Once you are done editing, scroll down and click the Save page button at the bottom of the page.

Please help improve this article by expanding it.
Further information might be found on the talk page or at requests for expansion.
This article has been tagged since May 2007.

Type-2 Gumbel
Probability density function
Cumulative distribution function
Parameters	$a\!$ (real) $b\!$ shape (real)
Support
Probability density function (pdf)	$a b x^{-a-1} \exp(-b x^{-a})\!$
Cumulative distribution function (cdf)	$\exp(-b x^{-a})\!$
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

In probability theory, the Type-2 Gumbel probability density function is

$f(x|a,b) = a b x^{-a-1} \exp(-b x^{-a})\,$

for

$0 < x < \infty$ .

This implies that it similar to the Weibull distributions, substituting $b = λ - k$ and $a = - k$ . Note however that a positive k (as in the Weibull distribution) would yield a negative a, which is not allowed here as it would yield a negative probability density.

For $0<a\le 1$ the mean is infinite. For $0<a\le 2$ the variance is infinite.

The cumulative distribution function is

$F(x|a,b) = \exp(-b x^{-a})\,$

The moments $E[X^k] \,$ exist for $k < a\,$

The special case b = 1 yelds the Fréchet distribution

Based on gsl-ref_19.html#SEC309, used under GFDL.

v • d • e

Probability distributions

[[Template: {{{name}}}\|v]] • [[Template talk: {{{name}}}\|d]] • [{{fullurl:Template: {{{name}}}\|action=edit}} e] Discrete univariate with finite support
Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

[[Template: {{{name}}}\|v]] • [[Template talk: {{{name}}}\|d]] • [{{fullurl:Template: {{{name}}}\|action=edit}} e] Discrete univariate with infinite support
Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

[[Template: {{{name}}}\|v]] • [[Template talk: {{{name}}}\|d]] • [{{fullurl:Template: {{{name}}}\|action=edit}} e] Continuous univariate supported on a bounded interval
Beta · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

[[Template: {{{name}}}|v]] • [[Template talk: {{{name}}}|d]] • [{{fullurl:Template: {{{name}}}|action=edit}} e]

Continuous univariate supported on a semi-infinite interval

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

[[Template: {{{name}}}\|v]] • [[Template talk: {{{name}}}\|d]] • [{{fullurl:Template: {{{name}}}\|action=edit}} e] Continuous univariate supported on the whole real line
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

[[Template: {{{name}}}\|v]] • [[Template talk: {{{name}}}\|d]] • [{{fullurl:Template: {{{name}}}\|action=edit}} e] Multivariate (joint)
Discrete: Ewens · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

[[Template: {{{name}}}\|v]] • [[Template talk: {{{name}}}\|d]] • [{{fullurl:Template: {{{name}}}\|action=edit}} e] Directional, degenerate, and singular
Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

[[Template: {{{name}}}\|v]] • [[Template talk: {{{name}}}\|d]] • [{{fullurl:Template: {{{name}}}\|action=edit}} e] Families
exponential · natural exponential · location-scale · maximum entropy · Pearson · Tweedie

Template:Probability-stub

Acknowledgement and Attribution Regarding Sources of Content

Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .

Retrieved from "http://www.wikidoc.org/index.php/Type-2_Gumbel_distribution"