|Probability mass function|
|Cumulative distribution function|
|Probability mass function (pmf)|
|Cumulative distribution function (cdf)|
|Moment-generating function (mgf)|
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 is given by:
which may be thought of as a generalization of a harmonic number. 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.
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).
- B. Mandelbrot (1965). "Information Theory and Psycholinguistics". In B.B. Wolman and E. Nagel. Scientific psychology. Basic Books. Reprinted as
- 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