probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, the Type-2 Gumbel

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

is :

f(x, a,b) = a b x^ e^\,

for :

0 < x < \infty

. For

0 the

mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithme ...

is infinite. For

0 the

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...

is infinite. The

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...

is :

F(x, a,b) = e^\,

The moments

E^k \,

exist for

k < a\,

The distribution is named after

Emil Julius Gumbel Emil Julius Gumbel (18 July 1891, in Munich – 10 September 1966, in New York City) was a German mathematician and political writer. Gumbel specialised in mathematical statistics and, along with Leonard Tippett and Ronald Fisher, was instrument ...

(1891 – 1966).

Generating random variates

Given a random variate ''U'' drawn from the

uniform distribution Uniform distribution may refer to: * Continuous uniform distribution * Discrete uniform distribution * Uniform distribution (ecology) * Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...

in the interval (0, 1), then the variate :

X=(-\ln U/b)^,

has a Type-2 Gumbel distribution with parameter

a

and

b

. This is obtained by applying the inverse transform sampling-method.

Related distributions

* The special case b = 1 yields the

Fréchet distribution The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function :\Pr(X \le x)=e^ \text x>0. where ''α'' > 0 is a ...

. * Substituting

b=\lambda^

and

a=-k

yields the Weibull distribution. Note, however, that a positive ''k'' (as in the Weibull distribution) would yield a negative ''a'' and hence a negative probability density, which is not allowed. ---- Based o
The GNU Scientific Library
used under GFDL.

Generating random variates

Related distributions

See also