In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
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
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
random variables.
The
Riemann zeta function being the sum of all terms
for positive integer ''k'', it appears thus as the normalization of the
Zipf distribution. 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
In probability and statistics, the Yule–Simon distribution is a discrete probability distribution named after Udny Yule and Herbert A. Simon. Simon originally called it the ''Yule distribution''.
The probability mass function (pmf) of the Yu ...
Definition
The Zeta distribution is defined for positive integers
, and its probability mass function is given by
:
,
where
is the parameter, and
is the
Riemann zeta function.
The cumulative distribution function is given by
:
where
is the generalized
harmonic number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers:
H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac.
Starting from , the sequence of harmonic numbers begins:
1, \frac, \frac, \frac, \frac, \dot ...
:
Moments
The ''n''th raw
moment is defined as the expected value of ''X''
''n'':
:
The series on the right is just a series representation of the Riemann zeta function, but it only converges for values of
that are greater than unity. Thus:
:
Note that the ratio of the zeta functions is well defined, even for ''n'' > ''s'' − 1 because the series representation of the zeta function can be
analytically continued
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ne ...
. This does not change the fact that the moments are specified by the series itself, and are therefore undefined for large ''n''.
Moment generating function
The
moment generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
is defined as
:
The series is just the definition of the
polylogarithm, valid for
so that
:
Since this does not converge on an open interval containing
, the moment generating function does not exist.
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
:
exists where ''N''(''A'', ''n'') is the number of members of ''A'' less than or equal to ''n'', then
:
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 proportional to
:
which is
Benford's law
Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.Arno Berger and Theodore ...
.
Infinite divisibility
The Zeta distribution can be constructed with a sequence of independent random variables with a
Geometric distribution
In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions:
* The probability distribution of the number ''X'' of Bernoulli trials needed to get one success, supported on the set \;
* ...
. Let
be a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
and
be a random variable with a Geometric distribution of parameter
, namely
If the random variables
are independent, then, the random variable
defined by
has the Zeta distribution :
.
Stated differently, the random variable
is
infinitely divisible
Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter, ...
with
Lévy measure
Levy, Lévy or Levies may refer to:
People
* Levy (surname), people with the surname Levy or Lévy
* Levy Adcock (born 1988), American football player
* Levy Barent Cohen (1747–1808), Dutch-born British financier and community worker
* Levy ...
given by the following sum of
Dirac masses :
See also
Other "power-law" distributions
*
Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
*
Lévy distribution
In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...
*
Lévy skew alpha-stable distribution
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be s ...
*
Pareto distribution
*
Zipf's law
*
Zipf–Mandelbrot law
In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution. Also known as the Pareto–Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf who sug ...
*
Infinitely divisible distribution In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables. The characteris ...
External links
* What Gut calls the "Riemann zeta distribution" is actually the probability distribution of −log ''X'', where ''X'' is a random variable with what this article calls the zeta distribution.
*
{{DEFAULTSORT:Zeta Distribution
Discrete distributions
Computational linguistics
Probability distributions with non-finite variance