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

			(4)
			(5)

The distribution has mean deviation

(6)

where  is a Hurwitz zeta function and  is the mean as given above in equation (4).