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Fisher-Tippett distribution
From Wikipedia, the free encyclopedia
Probability density function |
|
Cumulative distribution function |
|
| Parameters |  location (real) scale (real) |
|---|---|
| Support |  |
| Probability density function (pdf) |  where ![z = \exp\left[-\frac{x-\mu}{\beta}\right]\!](http://upload.wikimedia.org/math/3/9/3/39335168375a14882ef3d924cefd58fd.png) |
| Cumulative distribution function (cdf) | ![\exp(-\exp[-(x-\mu)/\beta])\!](http://upload.wikimedia.org/math/1/8/4/1849543b9e9c08b4c0c7bdef411e4e97.png) |
| Mean |  |
| Median |  |
| Mode | |
| Variance |  |
| Skewness |  |
| Excess kurtosis |  |
| Entropy |  for  |
| Moment-generating function (mgf) |  |
| Characteristic function |  |
In probability theory and statistics the Gumbel distribution (named after Emil Julius Gumbel (1891–1966)) is used to find the minimum (or the maximum) of a number of samples of various distributions. For example we would use it to find the maximum level of a river in a particular year if we had the list of maximum values for the past ten years. It is therefore useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur.
The distribution of the samples could be of the normal or exponential type. The Gumbel distribution, and similar distributions, are used in extreme value theory.
In particular, the Gumbel distribution is a special case of the Fisher-Tippett distribution (named after Sir Ronald Aylmer Fisher (1890–1962) and Leonard Henry Caleb Tippett (1902–1985)), also known as the log-Weibull distribution.
Contents |
The cumulative distribution function of the Fisher-Tippett distribution is
The median is μ − βln( − ln(0.5))
The mean is μ + γβ where γ = Euler-Mascheroni constant = 0.57721...
The standard deviation is
The mode is μ.
The standard Gumbel distribution is the case where μ = 0 and β = 1 with cumulative distribution function
and probability density function
The median is 
The mean is γ, the Euler-Mascheroni constant 0.57721...
The standard deviation is
The mode is 0.
A more practical way of using the distribution could be
where M is the median. To fit values one could get the median straight away and then vary μ until it fits the list of values.
Given a random variate U drawn from the uniform distribution in the interval (0, 1], the variate
has a Fisher-Tippett distribution with parameters μ and β. This follows from the form of the cumulative distribution function given above.
