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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

 P(x)=(x^(-(rho+1)))/(zeta(rho+1)),
(1)

where rho is a positive parameter and zeta(z) is the Riemann zeta function, and distribution function

 D(x)=(H_(x,rho+1))/(zeta(rho+1)),
(2)

where H_(n,r) is a generalized harmonic number.

The Zipf distribution is implemented in the Wolfram Language as ZipfDistribution[rho].

The nth raw moment is

 mu_n^'=(zeta(1-nrho))/(zeta(rho+1)),
(3)

giving the mean and variance as

mu=(zeta(rho))/(zeta(rho+1))
(4)
sigma^2=(zeta(rho-1))/(zeta(rho+1))-([zeta(rho)]^2)/([zeta(rho+1)]^2).
(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)),
(6)

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


See also

Zipf's Law

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Cite this as:

Weisstein, Eric W. "Zipf Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ZipfDistribution.html

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