# Zipf–Mandelbrot law

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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 Benoit Mandelbrot, who subsequently generalized it.

The probability mass function is given by:

where is given by:

which may be thought of as a generalization of a harmonic number. In the formula, is the rank of the data, and and are parameters of the distribution. In the limit as approaches infinity, this becomes the Hurwitz zeta function . For finite and the Zipf–Mandelbrot law becomes Zipf's law. For infinite and it becomes a Zeta distribution.

## Applications

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

If one plots the frequency rank of words contained in a moderately sized 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 Powers, 1998 and Gelbukh & Sidorov, 2001). Zipf's law implicitly assumes a fixed vocabulary size, but the Harmonic series with *s*=1 does not converge, while the Zipf-Mandelbrot generalization with *s*>1 does. Furthermore, there is evidence that the closed class of functional words that define a language obeys a Zipf-Mandelbrot distribution with different parameters from the open classes of contentive words that vary by topic, field and register.^{[1]}

In ecological field studies, the relative abundance distribution (i.e. the graph of the number of species observed as a function of their abundance) is often found to conform to a Zipf–Mandelbrot law.^{[2]}

Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandelbrot distributions.^{[3]}

## Notes

- ↑ Powers, David M W (1998). "Applications and explanations of Zipf's law". Association for Computational Linguistics: 151–160.
- ↑ Mouillot, D; Lepretre, A (2000). "Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity".
*Environmental Monitoring and Assessment*. Springer.**63**(2): 279–295. doi:10.1023/A:1006297211561. Retrieved 24 Dec 2008. - ↑ Manaris, B; Vaughan, D; Wagner, CS; Romero, J; Davis, RB. "Evolutionary Music and the Zipf-Mandelbrot Law: Developing Fitness Functions for Pleasant Music".
*Proceedings of 1st European Workshop on Evolutionary Music and Art (EvoMUSART2003)*.**611**.

## References

- Mandelbrot, Benoît (1965). "Information Theory and Psycholinguistics". In B.B. Wolman and E. Nagel.
*Scientific psychology*. Basic Books. Reprinted as- Mandelbrot, Benoît (1968) [1965]. "Information Theory and Psycholinguistics". In R.C. Oldfield and J.C. Marchall.
*Language*. Penguin Books.

- Mandelbrot, Benoît (1968) [1965]. "Information Theory and Psycholinguistics". In R.C. Oldfield and J.C. Marchall.
- Powers, David M W (1998). "Applications and explanations of Zipf's law". Association for Computational Linguistics: 151–160.
- Zipf, George Kingsley (1932).
*Selected Studies of the Principle of Relative Frequency in Language*. Cambridge, MA: Harvard University Press.