Zipf-Mandelbrot law

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Zipf-Mandelbrot
Probability mass function
Cumulative distribution function
Parameters	$N \in \{1,2,3\ldots\}$ (integer) $q \in [0;\infty)$ (real) $s>0\,$ (real)
Support	$k \in \{0,1,2,\ldots,N\}$
Probability mass function (pmf)	$\frac{1/(k+q)^s}{H_{N,q,s}}$
Cumulative distribution function (cdf)	$\frac{H_{k,q,s}}{H_{N,q,s}}$
Mean	$\frac{H_{N,q,s-1}}{H_{N,q,s}}-q$
Median	N/A
Mode	$\frac{1/(1+q)^s}{H_{N,q,s}}$
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 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:

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

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 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.

[edit] Applications

The distribution of words ranked by their frequency in a random corpus of writing 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).

[edit] 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

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