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.