The Zipf distribution, sometimes referred to as the zeta distribution, is a discrete distribution commonly used in linguistics, insurance, and the modelling of rare events. It has probability density function
(1)
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where is a positive parameter and is the Riemann zeta function, and distribution function
(2)
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where is a generalized harmonic number.
The Zipf distribution is implemented in the Wolfram Language as ZipfDistribution[rho].
The th raw moment is
(3)
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giving the mean and variance as
(4)
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(5)
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The distribution has mean deviation
(6)
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where is a Hurwitz zeta function and is the mean as given above in equation (4).