Zipf–Mandelbrot law

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.

Zipf–Mandelbrot

Parameters	${\displaystyle N\in \{1,2,3\ldots \}}$ (integer) ${\displaystyle q\in [0;\infty )}$ (real) ${\displaystyle s>0\,}$ (real)
Support	${\displaystyle k\in \{1,2,\ldots ,N\}}$
PMF	${\displaystyle {\frac {1/(k+q)^{s}}{H_{N,q,s}}}}$
CDF	${\displaystyle {\frac {H_{k,q,s}}{H_{N,q,s}}}}$
Mean	${\displaystyle {\frac {H_{N,q,s-1}}{H_{N,q,s}}}-q}$
Mode	${\displaystyle 1\,}$
Entropy	${\displaystyle {\frac {s}{H_{N,q,s}}}\sum _{k=1}^{N}{\frac {\ln(k+q)}{(k+q)^{s}}}+\ln(H_{N,q,s})}$

The probability mass function is given by:

${\displaystyle f(k;N,q,s)={\frac {1/(k+q)^{s}}{H_{N,q,s}}}}$

where ${\displaystyle H_{N,q,s}}$ is given by:

${\displaystyle H_{N,q,s}=\sum _{i=1}^{N}{\frac {1}{(i+q)^{s}}}}$

which may be thought of as a generalization of a harmonic number. In the formula, ${\displaystyle k}$ is the rank of the data, and ${\displaystyle q}$ and ${\displaystyle s}$ are parameters of the distribution. In the limit as ${\displaystyle N}$ approaches infinity, this becomes the Hurwitz zeta function ${\displaystyle \zeta (s,q)}$. For finite ${\displaystyle N}$ and ${\displaystyle q=0}$ the Zipf–Mandelbrot law becomes Zipf's law. For infinite ${\displaystyle N}$ and ${\displaystyle q=0}$ it becomes a Zeta distribution.