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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
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
s
∈
(
1
,
∞
)
{\displaystyle s\in (1,\infty )}
Support
k
∈
{
1
,
2
,
…
}
{\displaystyle k\in \{1,2,\ldots \}}
PMF
1
/
k
s
ζ
(
s
)
{\displaystyle {\frac {1/k^{s}}{\zeta (s )}}}
CDF
H
k
,
s
ζ
(
s
)
{\displaystyle {\frac {H_{k,s}}{\zeta (s )}}}
Mean
ζ
(
s
−
1
)
ζ
(
s
)
for
s
>
2
{\displaystyle {\frac {\zeta (s-1)}{\zeta (s )}}~{\textrm {for}}~s>2}
Mode
1
{\displaystyle 1\,}
Variance
ζ
(
s
)
ζ
(
s
−
2
)
−
ζ
(
s
−
1
)
2
ζ
(
s
)
2
for
s
>
3
{\displaystyle {\frac {\zeta (s )\zeta (s-2)-\zeta (s-1)^{2}}{\zeta (s )^{2}}}~{\textrm {for}}~s>3}
Entropy
∑
k
=
1
∞
1
/
k
s
ζ
(
s
)
log
(
k
s
ζ
(
s
)
)
.
{\displaystyle \sum _{k=1}^{\infty }{\frac {1/k^{s}}{\zeta (s )}}\log(k^{s}\zeta (s )).\,\!}
MGF
does not exist CF
Li
s
(
e
i
t
)
ζ
(
s
)
{\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
f
s
(
k
)
=
k
−
s
/
ζ
(
s
)
{\displaystyle f_{s}(k )=k^{-s}/\zeta (s )\,}
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
k
−
s
{\displaystyle k^{-s}}
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.
Definition [ edit ]
The Zeta distribution is defined for positive integers
k
≥
1
{\displaystyle k\geq 1}
, and its probability mass function is given by
P
(
x
=
k
)
=
1
ζ
(
s
)
k
−
s
{\displaystyle P(x=k)={\frac {1}{\zeta (s )}}k^{-s}}
,
where
s
>
1
{\displaystyle s>1}
is the parameter, and
ζ
(
s
)
{\displaystyle \zeta (s )}
is the Riemann zeta function .
The cumulative distribution function is given by
P
(
x
≤
k
)
=
H
k
,
s
ζ
(
s
)
,
{\displaystyle P(x\leq k)={\frac {H_{k,s}}{\zeta (s )}},}
where
H
k
,
s
{\displaystyle H_{k,s}}
is the generalized harmonic number
H
k
,
s
=
∑
i
=
1
k
1
i
s
.
{\displaystyle H_{k,s}=\sum _{i=1}^{k}{\frac {1}{i^{s}}}.}
Moments [ edit ]
The n th raw moment is defined as the expected value of X n :
m
n
=
E
(
X
n
)
=
1
ζ
(
s
)
∑
k
=
1
∞
1
k
s
−
n
{\displaystyle m_{n}=E(X^{n})={\frac {1}{\zeta (s )}}\sum _{k=1}^{\infty }{\frac {1}{k^{s-n}}}}
The series on the right is just a series representation of the Riemann zeta function, but it only converges for values of
s
−
n
{\displaystyle s-n}
that are greater than unity. Thus:
m
n
=
{
ζ
(
s
−
n
)
/
ζ
(
s
)
for
n
<
s
−
1
∞
for
n
≥
s
−
1
{\displaystyle m_{n}=\left\{{\begin{matrix}\zeta (s-n)/\zeta (s )&{\textrm {for}}~n<s-1\\\infty &{\textrm {for}}~n\geq s-1\end{matrix}}\right.}
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 [ edit ]
The moment generating function is defined as
M
(
t
;
s
)
=
E
(
e
t
X
)
=
1
ζ
(
s
)
∑
k
=
1
∞
e
t
k
k
s
.
{\displaystyle M(t;s)=E(e^{tX})={\frac {1}{\zeta (s )}}\sum _{k=1}^{\infty }{\frac {e^{tk}}{k^{s}}}.}
The series is just the definition of the polylogarithm , valid for
e
t
<
1
{\displaystyle e^{t}<1}
so that
M
(
t
;
s
)
=
Li
s
(
e
t
)
ζ
(
s
)
for
t
<
0.
{\displaystyle M(t;s)={\frac {\operatorname {Li} _{s}(e^{t})}{\zeta (s )}}{\text{ for }}t<0.}
Since this does not converge on an open interval containing
t
=
0
{\displaystyle t=0}
, the moment generating function does not exist.
The case s = 1 [ edit ]
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
lim
n
→
∞
N
(
A
,
n
)
n
{\displaystyle \lim _{n\to \infty }{\frac {N(A,n)}{n}}}
exists where N (A , n ) is the number of members of A less than or equal to n , then
lim
s
→
1
+
P
(
X
∈
A
)
{\displaystyle \lim _{s\to 1^{+}}P(X\in A)\,}
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
log
(
d
+
1
)
−
log
(
d
)
=
log
(
1
+
1
d
)
,
{\displaystyle \log(d+1)-\log(d )=\log \left(1+{\frac {1}{d}}\right),\,}
which is Benford's law .
Infinite divisibility [ edit ]
The Zeta distribution can be constructed with a sequence of independent random variables with a geometric distribution . Let
p
{\displaystyle p}
be a prime number and
X
(
p
−
s
)
{\displaystyle X(p^{-s})}
be a random variable with a geometric distribution of parameter
p
−
s
{\displaystyle p^{-s}}
, namely
P
(
X
(
p
−
s
)
=
k
)
=
p
−
k
s
(
1
−
p
−
s
)
{\displaystyle \quad \quad \quad \mathbb {P} \left(X(p^{-s})=k\right)=p^{-ks}(1-p^{-s})}
If the random variables
(
X
(
p
−
s
)
)
p
∈
P
{\displaystyle (X(p^{-s}))_{p\in {\mathcal {P}}}}
are independent, then, the random variable
Z
s
{\displaystyle Z_{s}}
defined by
Z
s
=
∏
p
∈
P
p
X
(
p
−
s
)
{\displaystyle \quad \quad \quad Z_{s}=\prod _{p\in {\mathcal {P}}}p^{X(p^{-s})}}
has the zeta distribution:
P
(
Z
s
=
n
)
=
1
n
s
ζ
(
s
)
{\displaystyle \mathbb {P} \left(Z_{s}=n\right)={\frac {1}{n^{s}\zeta (s )}}}
.
Stated differently, the random variable
log
(
Z
s
)
=
∑
p
∈
P
X
(
p
−
s
)
log
(
p
)
{\displaystyle \log(Z_{s})=\sum _{p\in {\mathcal {P}}}X(p^{-s})\,\log(p )}
is infinitely divisible with Lévy measure given by the following sum of Dirac masses :
Π
s
(
d
x
)
=
∑
p
∈
P
∑
k
⩾
1
p
−
k
s
k
δ
k
log
(
p
)
(
d
x
)
{\displaystyle \quad \quad \quad \Pi _{s}(dx )=\sum _{p\in {\mathcal {P}}}\sum _{k\geqslant 1}{\frac {p^{-ks}}{k}}\delta _{k\log(p )}(dx )}
See also [ edit ]
Other "power-law" distributions
External links [ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Zeta_distribution&oldid=1213415594 "
C a t e g o r i e s :
● D i s c r e t e d i s t r i b u t i o n s
● C o m p u t a t i o n a l l i n g u i s t i c s
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H i d d e n c a t e g o r i e s :
● A r t i c l e s n e e d i n g a d d i t i o n a l r e f e r e n c e s f r o m A u g u s t 2 0 1 1
● A l l a r t i c l e s n e e d i n g a d d i t i o n a l r e f e r e n c e s
● T h i s p a g e w a s l a s t e d i t e d o n 1 2 M a r c h 2 0 2 4 , a t 2 2 : 3 5 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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