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{{Probability distribution|
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  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
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:<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''&nbsp;&ge;&nbsp;''s''&nbsp;&minus;&nbsp;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'',&nbsp;''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 &minus;log&nbsp;''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]]

Revision as of 21:00, 9 February 2014

The best discounts are not available permanently. If an extraordinary offer is accessible, it will be offered for a although. Usually the cause why the deal is excellent is that somebody demands to get a offer completed in a restricted amount of time. And the individual who is ready to learn of this sort of a deal in time stands a great possibility of landing the outstanding deal. As a result when you are preparing to get or market real estate the quicker you can begin assessing real offer possibilities the far better use you will make of your time. Like others you too will have a window inside which to conclude the deal and the faster you commence obtaining info on true bargains the far better it will be for you.

When you update your kitchen by changing your previous countertop with a model new granite countertop, you are also updating the entire house. Even if you are only changing just the counter tops. Granite is costly and placing it on your kitchen is like an asset, raises the whole price tag of your property. Also, it significantly updates the complete kitchen which would also impact how the whole property appears. And if the complete property is current, then it would undoubtedly promote a great deal a lot more on the market.

Location normally takes up 60 % of all round house pleasure. When you're out buying for Jackson Gap homes for sale, appear for qualities at peaceful neighborhoods. After all, who would want to reside in an area with rampant criminal offense and violence? In addition, think about the proximity of your new house to educational institutions, hospitals, and to your perform. This must confirm valuable in moments of unexpected emergency.

If you do three houses for every calendar year, and you only Arthur Falcone net $25,000 total, following having to pay all expenses on each and every of the 3 residences, you are nevertheless netting $75,000 money and equity in about 6 to eight months. Furthermore, if you are leasing these homes, you are also making extra streams of cash flow by way of monthly funds stream as properly as accumulating equity in each and every home.

3 A 17. Assesses can make only one particular attractiveness against the purchase of the department to an Appellate Authority (Bench of 11 members). Buy of Appellate Authority ought to not be more Arthur Falcone appealable. Appellate Tribunals need to be setup in affordable numbers. Associates of tribunal should be nominated by professional bodies and the Govt. All appeals ought to be settled in maximum 3 hearings or a period of time of six months from the date of the filing of the attraction.

In basic, there are two sorts of real estate investments which consist of Speculative and Certainty. A speculative real estate expenditure has earnings that are reliant on appreciation. As a result, this variety of Arthur Falcone investing is susceptible to marketplace conditions and can make it a long expression selection.

Follow up instantly: If a vendor or purchaser asks you about some thing, make it a best precedence Arthur Falcone to get back again to them as fast as attainable. This will make your clientele feel important.

If you're views are already set for failure, effectively you're heading to fail. Sorry to start off unfavorable in this publish, but when push appear to shove, if that worry is your dominant considered, then it'll arrive to life.

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