Gumbel distribution

The Gumbel distribution (after Emil Julius Gumbel ), the Fisher - Tippett distribution ( by Ronald Aylmer Fisher) or extremal I- distribution is a continuous probability distribution that belongs as the Rossi - distribution and the Frechet distribution to extreme value distributions.

  • 2.1 Expectation value
  • 2.2 variance
  • 2.3 standard deviation
  • 4.1 Relationship to the extreme value distribution


A continuous random variable satisfies a Gumbel distribution with scale parameter and location parameter, if the probability density

And thus the distribution function


Standard case

If no parameters are specified, then the default parameters and mean. This results in the density

And the distribution function

The affine- linear transformations of the whole above mentioned class of distributions obtained with the properties


Expected value

The Gumbel distribution has the expected value

This is the Euler - Mascheroni constant.


The variance of a Gumbel distribution

Standard deviation

The standard deviation of a Gumbel distribution


It will, inter alia, used in the following areas:

  • Water for extreme events such as floods and droughts
  • Transport Planning
  • Meteorology ( Weather Forecast )
  • Hydrology

The Gumbel distribution is a typical distribution function for annual series. It can only be applied to rows, in which the length of the measurement series coincides with the sample size. Otherwise, you get negative logarithms.

Relationship to other distributions

Relationship with the extreme value distribution