Zeta distribution
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Template:Probability distribution
In probability theory and statistics, the zeta distribution is a discrete probability distribution. If <math>X<math> is a zeta-distributed random variable with parameter <math>s<math>, then the probability that <math>X<math> takes the integer value k is given by the probability mass function <math>f_k(s)<math>:
- <math>f_k(s)=k^{-s}/\zeta(s)\,<math>
where <math>\zeta(s)<math> is the Riemann zeta function.
It can be shown that these are the only probability distributions for which the multiplicities of distinct prime factors of X are independent random variables.
The zeta distribution is equivalent to the Zipf distribution for infinite N. Indeed the terms "Zipf distribution" and the "zeta distribution" are often used interchangeably.
The case s = 1
ζ(1) is infinite as the harmonic series, and so the case when s = 1 is not meaningful. However, if A is any set of positive integers that has a density, i.e. if
- <math>\lim_{n\rightarrow\infty}\frac{N(A,n)}{n}<math>
exists where N(A,n) is the number of members of A less than or equal to n, then
- <math>\lim_{s\rightarrow 1+}P(X\in A)\,<math>
is equal to that density.
The latter limit can also exist in some cases in which A does not have a density. For example, if A is the set of all positive integers whose first digit is d, then A has no density, but nonetheless the second limit given above exists and is equal to
- log10(d + 1) − log10(d),
in accord with Benford's law.
See also
Other "power-law" distributions
External links
- Some remarks on the Riemann zeta distributionby (http://www.math.uu.se/research/pub/Gut10.pdf) Allan Gut.de:Zipf-Verteilung