The distribution with probability density function and distribution function
(1)
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(2)
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defined over the interval .
It is implemented in the Wolfram Language as ParetoDistribution[k, alpha].
The th raw moment is
(3)
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for , giving the first few as
(4)
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(5)
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(6)
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(7)
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The th central moment is
(8)
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(9)
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for and where is a gamma function, is a regularized hypergeometric function, and is a beta function, giving the first few as
(10)
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(11)
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(12)
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The mean, variance, skewness, and kurtosis excess are therefore
(13)
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(14)
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(15)
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(16)
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