Zipf-Mandelbrot law
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cdf =<math>\frac{H_{k,q,s}}{H_{N,q,s}}</math>|
mean =<math>\frac{H_{N,q,s-1}}{H_{N,q,s}}-q</math>|
median =N/A|
mode =<math>\frac{1/(1+q)^s}{H_{N,q,s}}</math>|
variance =|
skewness =|
kurtosis =|
entropy =|
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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 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:
- <math>f(k;N,q,s)=\frac{1/(k+q)^s}{H_{N,q,s}}</math>
where <math>H_{N,q,s}</math> is given by:
- <math>H_{N,q,s}=\sum_{i=1}^N \frac{1}{(i+q)^s}</math>
which may be thought of as a generalization of a harmonic number. In the limit as <math>N</math> approaches infinity, this becomes the Hurwitz zeta function <math>\zeta(q,s)</math>. For finite <math>N</math> and <math>q=0</math> the Zipf-Mandelbrot law becomes Zipf's law. For infinite <math>N</math> and <math>q=0</math> 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).