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Extreme value distributions_Weibull

Extreme value distributions_Weibull - Extreme value...

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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 well-behaved 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
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