|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Probability distribution|
| | They contact me Emilia. The favorite hobby for my kids and me is to play baseball and I'm attempting to make it a profession. Minnesota is where he's been living for years. For many years he's been working as a receptionist.<br><br>my web-site :: [http://www.quarterpathtrace.com/members/marcivee/activity/58056/ http://www.quarterpathtrace.com/members/marcivee/activity/58056/] |
| name =zeta|
| |
| type =mass|
| |
| pdf_image =[[Image:Zeta distribution PMF.png|325px|Plot of the Zeta PMF]]<br /><small>Plot of the Zeta PMF on a log-log scale. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)</small>|
| |
| cdf_image =[[Image:Zeta distribution CMF.png|325px|Plot of the Zeta CMF]]|
| |
| parameters =<math>s\in(1,\infty)</math>|
| |
| support =<math>k \in \{1,2,\ldots\}</math>|
| |
| pdf =<math>\frac{1/k^s}{\zeta(s)}</math>|
| |
| cdf =<math>\frac{H_{k,s}}{\zeta(s)}</math>|
| |
| mean =<math>\frac{\zeta(s-1)}{\zeta(s)}~\textrm{for}~s>2</math>|
| |
| median =|
| |
| mode =<math>1\,</math>|
| |
| variance =<math>\frac{\zeta(s)\zeta(s-2) - \zeta(s-1)^2}{\zeta(s)^2}~\textrm{for}~s>3</math>|
| |
| skewness =|
| |
| kurtosis =|
| |
| entropy =<math>\sum_{k=1}^\infty\frac{1/k^s}{\zeta(s)}\log (k^s \zeta(s)).\,\!</math>|
| |
| mgf =<math>\frac{\operatorname{Li}_s(e^t)}{\zeta(s)}</math>|
| |
| char =<math>\frac{\operatorname{Li}_s(e^{it})}{\zeta(s)}</math>|
| |
| }}
| |
| | |
| 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
| |
| | |
| :<math>f_s(k)=k^{-s}/\zeta(s)\,</math>
| |
| | |
| where ζ(''s'') is the [[Riemann zeta function]] (which is undefined for ''s'' = 1). | |
| | |
| The multiplicities of distinct [[prime factor]]s of ''X'' are [[statistical independence|independent]] [[random variable]]s.
| |
| | |
| The zeta distribution is equivalent to the [[Zipf's law|Zipf distribution]] for infinite ''N''{{Elucidate|date=December 2012}}. Indeed the terms "Zipf distribution" and the "zeta distribution" are often used interchangeably.
| |
| | |
| == Moments ==
| |
| | |
| The ''n''th raw [[moment (mathematics)|moment]] is defined as the expected value of ''X''<sup>''n''</sup>:
| |
| | |
| :<math>m_n = E(X^n) = \frac{1}{\zeta(s)}\sum_{k=1}^\infty \frac{1}{k^{s-n}}</math>
| |
| | |
| The series on the right is just a series representation of the Riemann zeta function, but it only converges for values of ''s-n'' that are greater than unity. Thus:
| |
| | |
| :<math>m_n =\left\{ | |
| \begin{matrix}
| |
| \zeta(s-n)/\zeta(s) & \textrm{for}~n < s-1 \\
| |
| \infty & \textrm{for}~n \ge s-1
| |
| \end{matrix}
| |
| \right.
| |
| </math>
| |
| | |
| 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 [[analytic continuation|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
| |
| | |
| :<math>M(t;s) = E(e^{tX}) = \frac{1}{\zeta(s)} \sum_{k=1}^\infty \frac{e^{tk}}{k^s}.</math> | |
| | |
| The series is just the definition of the [[polylogarithm]], valid for <math>e^t<1</math> so that
| |
| | |
| :<math>M(t;s) = \frac{\operatorname{Li}_s(e^t)}{\zeta(s)}\text{ for }t<0.</math>
| |
| | |
| The [[Taylor series]] expansion of this function will not necessarily yield the moments of the distribution. The Taylor series using the moments as they usually occur in the moment generating function yields
| |
| | |
| :<math>\sum_{n=0}^\infty \frac{m_n t^n}{n!},</math>
| |
| | |
| which obviously is not well defined for any finite value of ''s'' since the moments become infinite for large ''n''. If we use the analytically continued terms instead of the moments themselves, we obtain from a series representation of the [[polylogarithm]]
| |
| | |
| :<math>\frac{1}{\zeta(s)}\sum_{n=0,n\ne s-1}^\infty \frac{\zeta(s-n)}{n!}\,t^n=\frac{\operatorname{Li}_s(e^t)-\Phi(s,t)}{\zeta(s)}</math>
| |
| | |
| for <math>\scriptstyle |t|\,<\,2\pi</math>. <math>\scriptstyle\Phi(s,t)</math> is given by
| |
| | |
| :<math>\Phi(s,t)=\Gamma(1-s)(-t)^{s-1}\text{ for }s\ne 1,2,3\ldots</math>
| |
| :<math>\Phi(s,t)=\frac{t^{s-1}}{(s-1)!}\left[H_s-\ln(-t)\right]\text{ for }s=2,3,4\ldots</math>
| |
| :<math>\Phi(s,t)=-\ln(-t)\text{ for }s=1,\,</math> | |
| | |
| where ''H''<sub>''s''</sub> is a [[harmonic number]].
| |
| | |
| ==The case ''s'' = 1==
| |
| | |
| ζ(1) is infinite as the [[harmonic series (mathematics)|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 proportional to
| |
| | |
| :<math>\log(d+1) - \log(d),\,</math>
| |
| | |
| similar to [[Benford's law]].
| |
| | |
| == See also ==
| |
| | |
| Other "power-law" distributions
| |
| | |
| *[[Cauchy distribution]]
| |
| *[[Lévy distribution]]
| |
| *[[Lévy skew alpha-stable distribution]]
| |
| *[[Pareto distribution]]
| |
| *[[Zipf's law]]
| |
| *[[Zipf–Mandelbrot law]]
| |
| | |
| {{unreferenced|date=August 2011}}
| |
| | |
| == External links ==
| |
| | |
| * ''[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.66.3284&rep=rep1&type=pdf Some remarks on the Riemann zeta distribution]'' by Allan Gut. 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.
| |
| * {{MathWorld |name=Zipf Distribution |id=ZipfDistribution}}
| |
| | |
| {{ProbDistributions|Zeta distribution}}
| |
| | |
| {{DEFAULTSORT:Zeta Distribution}}
| |
| [[Category:Discrete distributions]]
| |
| [[Category:Computational linguistics]]
| |
| [[Category:Probability distributions with non-finite variance]]
| |
| [[Category:Probability distributions]]
| |
They contact me Emilia. The favorite hobby for my kids and me is to play baseball and I'm attempting to make it a profession. Minnesota is where he's been living for years. For many years he's been working as a receptionist.
my web-site :: http://www.quarterpathtrace.com/members/marcivee/activity/58056/