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{{Probability distribution |
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  name      =Type-2 Gumbel|
  type      =density|
  pdf_image =|
  cdf_image =|
   parameters =<math>a\!</math> ([[real number|real]])<br /><math>b\!</math> shape (real)|
  support    =|
  pdf        =<math> a b x^{-a-1} e^{-b x^{-a}}\!</math>|
  cdf        =<math> e^{-b x^{-a}}\!</math>|
  mean      =|
  median    =|
  mode      =|
  variance  =|
  skewness  =|
  kurtosis  =|
  entropy   =|
  mgf        =|
  char      =|
}}
In [[probability theory]], the '''Type-2 Gumbel [[probability density function]]''' is
 
:<math>f(x|a,b) = a b x^{-a-1} e^{-b x^{-a}}\,</math>
 
for
 
:<math>0 < x < \infty</math>.
 
This implies that it is similar to the [[Weibull distribution]]s, substituting <math>b=\lambda^{-k}</math> and <math>a=-k</math>. 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 <math>0<a\le 1</math> the [[mean]] is infinite. For <math>0<a\le 2</math> the [[variance]] is infinite.
 
The [[cumulative distribution function]] is
 
:<math>F(x|a,b) = e^{-b x^{-a}}\,</math>
 
The moments <math> E[X^k] \,</math> exist for <math>k < a\,</math>
 
The special case b = 1 yields the [[Fréchet distribution]]
 
----
Based on [http://www.gnu.org/software/gsl/manual/html_node/The-Type_002d2-Gumbel-Distribution.html The GNU Scientific Library], used under GFDL.
 
==See also==
* [[Extreme value theory]]
* [[Gumbel distribution]]
* [[Type-1 Gumbel distribution]]
 
{{ProbDistributions|continuous-semi-infinite}}
 
[[Category:Continuous distributions]]
[[Category:Probability distributions]]

Revision as of 12:49, 18 February 2014

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