# Zipf-Mandelbrot law

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Parameters Probability mass function Cumulative distribution function $N\in \{1,2,3\ldots \}$ (integer)$q\in [0;\infty )$ (real)$s>0\,$ (real) $k\in \{1,2,\ldots ,N\}$ ${\frac {1/(k+q)^{s}}{H_{N,q,s}}}$ ${\frac {H_{k,q,s}}{H_{N,q,s}}}$ ${\frac {H_{N,q,s-1}}{H_{N,q,s}}}-q$ $1\,$ 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 $\zeta (q,s)$ . 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 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). 