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
Definition
A continuous random variable satisfies a Gumbel distribution with scale parameter and location parameter, if the probability density
And thus the distribution function
Possesses.
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
Properties
Expected value
The Gumbel distribution has the expected value
This is the Euler - Mascheroni constant.
Variance
The variance of a Gumbel distribution
Standard deviation
The standard deviation of a Gumbel distribution
Application
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.