# 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 length distribution of 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*_{n}, the *n*-th harmonic number.

A Zipf-like distribution with parameters *n* and *s* assigns probability proportional to 1/*r*^{s} to an integer *r* ≤ *n* and zero otherwise, with normalization factor .

The zeta distribution with parameter *s*>1 assigns probability proportional to 1/*r*^{s} to all integers *r* with normalization factor given by 1/ζ(*s*) where ζ(*s*) is the Riemann zeta function .

## [edit] References

- Michael Woodroofe; Bruce Hill (1975).
*On Zipf's law*, 425-434. Zbl 0343.60012.

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