# Zipf–Mandelbrot law

## From Wikipedia, the free encyclopedia

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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]}

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