# Zipf–Mandelbrot law

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 Probability mass function Cumulative distribution function Parameters $N \in \{1,2,3\ldots\}$ (integer)$q \in [0;\infty)$ (real)$s>0\,$ (real) Support $k \in \{1,2,\ldots,N\}$ Template:Probability distribution/link mass $\frac{1/(k+q)^s}{H_{N,q,s}}$ cdf $\frac{H_{k,q,s}}{H_{N,q,s}}$ Mean $\frac{H_{N,q,s-1}}{H_{N,q,s}}-q$ Median Mode $1\,$ Variance Skewness Kurtosis Entropy mgf Char. func.

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:

$f(k;N,q,s)=\frac{1/(k+q)^s}{H_{N,q,s}}$

where $H_{N,q,s}$ is given by:

$H_{N,q,s}=\sum_{i=1}^N \frac{1}{(i+q)^s}$

which may be thought of as a generalization of a harmonic number. In the formula, k is the rank of the data, and q and s are parameters of the distribution. In the limit as $N$ approaches infinity, this becomes the Hurwitz zeta function $\zeta(q,s)$. 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.

## ApplicationsEdit

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 & Sidorov, 2001).

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.[1]

Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandlebrot distributions.[2]

## NotesEdit

1. 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 63 (2): 279–295.
2. Manris, 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.

## ReferencesEdit

• Mandelbrot, Benoît (1965). "Information Theory and Psycholinguistics" B.B. Wolman and E. Nagel Scientific psychology, Basic Books. Reprinted as
• Mandelbrot, Benoît [1965] (1968). "Information Theory and Psycholinguistics" R.C. Oldfield and J.C. Marchall Language, Penguin Books.
• Zipf, George Kingsley (1932). Selected Studies of the Principle of Relative Frequency in Language, Cambridge, MA: Harvard University Press.