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| Probability mass function |
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| Cumulative distribution function |
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| 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 linguist George Kingsley Zipf who suggested a simpler distribution called Zipf's law, and the mathematician Benoît Mandelbrot, who subsequently generalized it.
The probability mass function is given by:
where HN,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 text corpus 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
