Zipf–Mandelbrot law
</math>| 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 formula_6 is given by: which may be thought of as a generalization of a harmonic number. In the formula, formula_8 is the rank of the data, and formula_9 and formula_10 are parameters  of the distribution. In the limit as formula_11 approaches infinity, this becomes the Hurwitz zeta function formula_12. For finite formula_11 and formula_14 the Zipf–Mandelbrot law becomes Zipf's law. For infinite formula_11 and formula_14 it becomes a Zeta distribution. Applications. The distribution of words ranked by their frequency in a random text corpus is approximated by a power-law distribution, known as Zipf's law. If one plots the frequency rank of words contained in a moderately sized 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 Powers, 1998 and Gelbukh & Sidorov, 2001). Zipf's law implicitly assumes a fixed vocabulary size, but the Harmonic series with "s"=1 does not converge, while the Zipf–Mandelbrot generalization with "s">1 does. Furthermore, there is evidence that the closed class of functional words that define a language obeys a Zipf–Mandelbrot distribution with different parameters from the open classes of contentive words that vary by topic, field and register. In ecological field studies, the relative abundance distribution (i.e. the graph of the number of species observed as a function of their abundance) is often found to  conform to a Zipf–Mandelbrot law. Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandelbrot distributions.