The Weibull distribution is given by
(1)
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(2)
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for , and is implemented in the Wolfram Language as WeibullDistribution[alpha, beta]. The raw moments of the distribution are
(3)
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(4)
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(5)
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(6)
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and the mean, variance, skewness, and kurtosis excess of are
(7)
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(8)
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(9)
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(10)
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where is the gamma function and
(11)
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A slightly different form of the distribution is defined by
(12)
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(13)
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(Mendenhall and Sincich 1995). This has raw moments
(14)
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(15)
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(16)
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(17)
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so the mean and variance for this form are
(18)
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(19)
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The Weibull distribution gives the distribution of lifetimes of objects. It was originally proposed to quantify fatigue data, but it is also used in analysis of systems involving a "weakest link."