Zeta distribution

Zeta distribution with different parameter values ​​from s

The zeta distribution (also Zipf distribution according to George Kingsley Zipf ) is a probability distribution in stochastics . It is univariate and a discrete probability distribution that gives the natural numbers the probabilities ${\displaystyle x=1,2,3,\dotsc }$

${\displaystyle P(X=x)={\frac {x^{-s}}{\zeta (s)}}}$

assigns, where is a parameter and the Riemann zeta function . ${\displaystyle s>1}$ ${\displaystyle \zeta (s)}$

Your th moment exists, if , and is enclosed in this case ${\displaystyle k}$ ${\displaystyle s>k+1}$

${\displaystyle E(X^{k})={\frac {\zeta (s-k)}{\zeta (s)}}}$.

It can be shown that the number of different prime factors of a zeta-distributed random variable are in turn independent random variables. This is not the case with any other probability distribution.

See Zipf's law for the motivation for this distribution .

Web links

• Eric W. Weisstein : Zipf distribution . In: MathWorld (English).
• The size of US companies obeys the mathematical Zipf distribution on Wissenschaft.de