Extreme value distributions
•
Description, Formulas and Plots
•
Uses of the Extreme Value Distribution Model
•
DATAPLOT Functions for the Extreme Value Distribution
The Extreme
Value
Distribution
usually refers to
the distribution
of the minimum
of a large
number of
unbounded
random
observations
Description, Formulas and Plots
We have already referred to
Extreme Value Distributions
when describing the
uses of the
Weibull distribution
. Extreme value distributions are the limiting distributions for the
minimum or the maximum of a very large collection of random observations from the same
arbitrary distribution. Gumbel (1958) showed that for any wellbehaved initial distribution
(i.e.,
F(x)
is continuous and has an inverse), only a few models are needed, depending on
whether you are interested in the maximum or the minimum, and also if the observations are
bounded above or below.
In the context of reliability modeling, extreme value distributions for the minimum are
frequently encountered. For example, if a system consists of
n
identical components in series,
and the system fails when the first of these components fails, then system failure times are the
minimum of
n
random component failure times. Extreme value theory says that, independent
of the choice of component model, the system model will approach a Weibull as
n
becomes
large. The same reasoning can also be applied at a component level, if the component failure
occurs when the first of many similar competing failure processes reaches a critical level.
The distribution often referred to as the
Extreme Value Distribution (Type I)
is the limiting
distribution of the minimum of a large number of unbounded identically distributed random
variables. The PDF and CDF are given by:
Extreme Value
Distribution
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '08
 Staff
 Normal Distribution, Probability theory, probability density function, Cumulative distribution function, Extreme value, Gumbel

Click to edit the document details