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
 |  |  |
(4)
|
 |  | ![(zeta(rho-1))/(zeta(rho+1))-([zeta(rho)]^2)/([zeta(rho+1)]^2).](/images/equations/ZipfDistribution/Inline10.svg) |
(5)
|
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)),](/images/equations/ZipfDistribution/NumberedEquation4.svg) |
(6)
|
where 
is a Hurwitz zeta function and 
is the mean as given above in equation (4).
