Zipf-Mandelbrot law

Zipf-Mandelbrot

Probability mass function
Cumulative distribution function
Parameters	(integer) (real) (real)
Support
Probability mass function (pmf)
Cumulative distribution function (cdf)
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. Also known as the Pareto-Zipf law, it is a power-law distribution on ranked data, named after the Harvard linguistics professor George Kingsley Zipf (1902-1950) who suggested a simpler distribution called Zipf's law, and the mathematician Benoît Mandelbrot (born November 20, 1924), who subsequently generalized it.

The probability mass function is given by:

where H_N,q,s is given by:

which may be thought of as a generalization of a harmonic number. In the limit as N approaches infinity, this becomes the Hurwitz zeta function ζ(q,s). For finite N and q = 0 the Zipf-Mandelbrot law becomes Zipf's law. For infinite N and q = 0 it becomes a Zeta distribution.

Applications

The distribution of words ranked by their frequency in a random corpus of writing is generally a power-law distribution, known as Zipf's law.

If one plots the frequency rank of words contained in a large corpus of text data versus the number of occurrences or actual frequencies, one obtains a power-law distribution, with exponent close to one (but see Gelbukh and Sidorov 2001).

References and links

• B. Mandelbrot (1965). "Information Theory and Psycholinguistics", in B.B. Wolman and E. Nagel: Scientific psychology. Basic Books.  Reprinted as
  • B. Mandelbrot [1965] (1968). "Information Theory and Psycholinguistics", in R.C. Oldfield and J.C. Marchall: Language. Penguin Books. 
• Z. K. Silagadze: Citations and the Zipf-Mandelbrot's law
• NIST: Zipf's law
• W. Li's References on Zipf's law
• Gelbukh and Sidorov 2001: Zipf and Heaps Laws’ Coefficients Depend on Language

	Probability distributions []
	Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial • multivariate Polya
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • Generalized Dirichlet distribution . inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	bimodal • Cantor • conditional • equilibrium • exponential family • Infinite divisibility (probability) • location-scale family • marginal • maximum entropy • posterior • prior • quasi • sampling • singular

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