Difference between revisions of "User:Richard Pinch/sandbox-CZ"
(→Wiener–Ikehara theorem: transfer) |
(Start article: Zipf distribution) |
||
| Line 1: | Line 1: | ||
| + | =Zipf distribution= | ||
| + | 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]]. | ||
| + | |||
| + | A Zipf-like distribution with parameters ''n'' and ''s'' assigns probability proportional to 1/''r''<sup>''s''</sup> to an integer ''r'' ≤ ''n'' and zero otherwise, with normalization factor <math>\sum_{r=1}^n 1/r^s</math>. | ||
| + | |||
| + | 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 06:39, 8 September 2013
Zipf distribution
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 \(\sum_{r=1}^n 1/r^s\).
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
Richard Pinch/sandbox-CZ. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-CZ&oldid=30415