Type-2 Gumbel distribution

Type-2 Gumbel
Parameters	$a\!$ (real) $b\!$ shape (real)
PDF	$a b x^{-a-1} e^{-b x^{-a}}\!$
CDF	$e^{-b x^{-a}}\!$

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

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

for

0 < x < \infty

This implies that it is similar to the Weibull distributions, substituting $b=\lambda^{-k}$ 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) = e^{-b x^{-a}}\,

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

The special case b = 1 yields the Fréchet distribution

Based on The GNU Scientific Library, used under GFDL.

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace Asymmetric Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Type-2 Gumbel distribution

See also