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The Weibull fit was originally used because of a belief that particle energy levels align to a statistical distribution, but this belief was later proven false [citation needed] and the Weibull fit continues to be used because of its many adjustable parameters, rather than a demonstrated physical basis.
It is characterized by a single parameter, λ, which is both the mean and variance of the distribution. The discrete Weibull distribution, on the other hand, is more flexible and can handle both over- and under-dispersion in count data. It has two parameters, q and β, which influence the shape and scale of the distribution.
This arises because the ordinary Weibull distribution is used for cases that deal with data minima rather than data maxima. The distribution here has an addition parameter compared to the usual form of the Weibull distribution and, in addition, is reversed so that the distribution has an upper bound rather than a lower bound.
The Fréchet distribution, also known as inverse Weibull distribution, [2] [3] is a special case of the generalized extreme value distribution. It has the cumulative distribution function ( ) = > . where α > 0 is a shape parameter.
[1] is a parameter found during the fit of data to the Weibull distribution and represents an input value for which ~67% of the data is encompassed. As m increases, the CDF distribution more closely resembles a step function at x 0 {\displaystyle x_{0}} , which correlates with a sharper peak in the probability density function (PDF) defined by:
In statistics, the exponentiated Weibull family of probability distributions was introduced by Mudholkar and Srivastava (1993) as an extension of the Weibull family obtained by adding a second shape parameter. The cumulative distribution function for the exponentiated Weibull distribution is
The Weibull distribution with the shape parameter k = 2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter σ {\displaystyle \sigma } is related to the Weibull scale parameter according to λ = σ 2 . {\displaystyle \lambda =\sigma {\sqrt {2}}.}
Horizontal axis: Value of shape parameter. That is, for a series of values of the shape parameter, the correlation coefficient is computed for the probability plot associated with a given value of the shape parameter. These correlation coefficients are plotted against their corresponding shape parameters.