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 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 $\sum _{r=1}^{n}1/r^{s}$.

The zeta distribution with parameter s assigns probability proportional to 1/r^s 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.