In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, the Type-2 Gumbel probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

is


for
.

This implies that it is similar to the Weibull distributions, substituting and . 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 the mean

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

 is infinite. For the variance

Variance

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

 is infinite.

The cumulative distribution function

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

 is


The moments exist for

The special case b = 1 yields the Fréchet distribution

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Based on The GNU Scientific Library, used under GFDL.

See also

• Extreme value theory
  Extreme value theory
  Extreme value theory is a branch of statistics dealing with the extreme deviations from the median of probability distributions. The general theory sets out to assess the type of probability distributions generated by processes...
  
  
• Gumbel distribution
• Type-1 Gumbel distribution