Zeta distribution
None
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.) |
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Cumulative distribution function
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Parameters |
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Support |
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PMF |
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CDF |
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Mean |
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Mode |
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Variance |
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Entropy |
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MGF | does not exist |
CF |
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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
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 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
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
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:
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 . 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 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 .
Infinite divisibility
The Zeta distribution can be constructed with a sequence of independent random variables with a
Geometric distribution
. Let
be a
prime number
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
with
Lévy measure
given by the following sum of
Dirac masses
:
See also
Other "power-law" distributions
External links
- Gut, Allan. "Some remarks on the Riemann zeta distribution". CiteSeerX 10.1.1.66.3284 . 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.
- Weisstein, Eric W. "Zipf Distribution" . MathWorld .