Zipf–Mandelbrot law
Discrete probability distribution / 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.