Zipf Distribution
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19-4-2021
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Zipf Distribution
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)
|
where 
is a positive parameter and 
is the Riemann zeta function, and distribution function
 |
(2)
|
where 
is a generalized harmonic number.
The Zipf distribution is implemented in the Wolfram Language as ZipfDistribution[rho].
The 
th raw moment is
 |
(3)
|
giving the mean and variance as
The distribution has mean deviation
![MD=(2[zeta(rho+1)zeta(rho,|_mu_|+1)-zeta(rho)zeta(rho+1,|_mu_|+1)])/(zeta^2(rho+1)),](https://mathworld.wolfram.com/images/equations/ZipfDistribution/NumberedEquation4.gif) |
(6)
|
where 
is a Hurwitz zeta function and 
is the mean as given above in equation (4).
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