Zeta distribution

zeta

Probability mass function Plot of the Zeta PMF on a log-log scale. (The function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Cumulative distribution function
Parameters	${\displaystyle s\in (1,\infty )}$
Support	${\displaystyle k\in \{1,2,\ldots \}}$
pmf	${\displaystyle {\frac {1/k^{s}}{\zeta (s)}}}$
CDF	${\displaystyle {\frac {H_{k,s}}{\zeta (s)}}}$
Mean	${\displaystyle {\frac {\zeta (s-1)}{\zeta (s)}}~{\textrm {for}}~s>2}$
Mode	${\displaystyle 1\,}$
Variance	${\displaystyle {\frac {\zeta (s)\zeta (s-2)-\zeta (s-1)^{2}}{\zeta (s)^{2}}}~{\textrm {for}}~s>3}$
Entropy	${\displaystyle \sum _{k=1}^{\infty }{\frac {1/k^{s}}{\zeta (s)}}\log(k^{s}\zeta (s)).\,\!}$
MGF	${\displaystyle {\frac {\operatorname {Li} _{s}(e^{t})}{\zeta (s)}}}$
CF	${\displaystyle {\frac {\operatorname {Li} _{s}(e^{it})}{\zeta (s)}}}$

In probability theory and statistics, the zeta distribution is a discrete probability distribution. If X is a zeta-distributed random variable with parameter s, then the probability that X takes the integer value k is given by the probability mass function

where ζ(s) is the Riemann zeta function (which is undefined for s = 1).

The multiplicities of distinct prime factors of X are independent random variables.

The Riemann zeta function being the sum of all terms ${\displaystyle k^{-s}}$ for positive integer k, it appears thus as the normalization of the Zipf distribution. Indeed the terms "Zipf distribution" and the "zeta distribution" are often used interchangeably. But note that while the Zeta distribution is a probability distribution by itself, it is not associated to the Zipf's law with same exponent. See also Yule–Simon distribution

The nth raw moment is defined as the expected value of Xⁿ: