Zipf-Mandelbrot law

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]| cdf =[\frac}}]| mean =[\frac}}-q]| median =N/A| mode =[\frac}]| variance =| skewness =| kurtosis =| entropy =| mgf =| char =| }} 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 Benoit Mandelbrot (born November 20, 1924), who subsequently generalized it.

The probability mass function is given by:

[f(k;N,q,s)=\frac}]

where [H_] is given by:

[H_=\sum_^N \frac]

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 [\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.

Applications

The distribution of words ranked by their frequency in a random corpus of text 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 Sidoro 2001).

External links

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