Zipf distribution: Difference between revisions
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In [[probability theory]] and [[statistics]], the '''Zipf distribution''' and '''zeta distribution''' refer to a class of [[discrete probability distribution]]s. They have been used to model the distribution of text strings and keys in databases. | In [[probability theory]] and [[statistics]], the '''Zipf distribution''' and '''zeta distribution''' refer to a class of [[discrete probability distribution]]s. They have been used to model the distribution of words in words in a text , of text strings and keys in databases, and of the sizes of businesses and towns. | ||
The Zipf distribution with parameter ''n'' assigns probability proportional to 1/''r'' to an integer ''r'' ≤ ''n'' and zero otherwise, with [[normalization]] factor ''H''<sub>''n''</sub>, the ''n''-th [[harmonic number]]. | The Zipf distribution with parameter ''n'' assigns probability proportional to 1/''r'' to an integer ''r'' ≤ ''n'' and zero otherwise, with [[normalization]] factor ''H''<sub>''n''</sub>, the ''n''-th [[harmonic number]]. | ||
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The zeta distribution with parameter ''s'' assigns probability proportional to 1/''r''<sup>''s''</sup> to all integers ''r'' with normalization factor given by the [[Riemann zeta function]] 1/ζ(''s''). | The zeta distribution with parameter ''s'' assigns probability proportional to 1/''r''<sup>''s''</sup> to all integers ''r'' with normalization factor given by the [[Riemann zeta function]] 1/ζ(''s''). | ||
==References== | |||
* {{cite book | author=Michael Woodroofe | coauthors=Bruce Hill | title=On Zipf's law | journal=J. Appl. Probab. | volume=12 | pages=425-434 | year=1975 | id=Zbl 0343.60012 }} | |||
Revision as of 14:54, 21 December 2008
In probability theory and statistics, the Zipf distribution and zeta distribution refer to a class of discrete probability distributions. They have been used to model the distribution of words in words in a text , of text strings and keys in databases, and of the sizes of businesses and towns.
The Zipf distribution with parameter n assigns probability proportional to 1/r to an integer r ≤ n and zero otherwise, with normalization factor Hn, the n-th harmonic number.
A Zipf-like distribution with parameters n and s assigns probability proportional to 1/rs to an integer r ≤ n and zero otherwise, with normalization factor .
The zeta distribution with parameter s assigns probability proportional to 1/rs to all integers r with normalization factor given by the Riemann zeta function 1/ζ(s).
References
- Michael Woodroofe; Bruce Hill (1975). On Zipf's law, 425-434. Zbl 0343.60012.