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| <!-- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion
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| of standards used for probability distribution articles such as this one. -->
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| {{Probability distribution|
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| name =Zipf–Mandelbrot|
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| type =mass|
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| pdf_image =|
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| cdf_image =|
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| parameters =<math>N \in \{1,2,3\ldots\}</math> ([[integer]])<br /><math>q \in [0;\infty)</math> ([[Real number|real]])<br /><math>s>0\,</math> ([[real number|real]])|
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| support =<math>k \in \{1,2,\ldots,N\}</math>|
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| pdf =<math>\frac{1/(k+q)^s}{H_{N,q,s}}</math>|
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| cdf =<math>\frac{H_{k,q,s}}{H_{N,q,s}}</math>|
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| mean =<math>\frac{H_{N,q,s-1}}{H_{N,q,s}}-q</math>|
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| median =|
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| mode =<math>1\,</math>|
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| variance =|
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| skewness =|
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| kurtosis =|
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| entropy =|
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| mgf =|
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| char =|
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| }}
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| In [[probability theory]] and [[statistics]], the '''Zipf–Mandelbrot law''' is a [[discrete probability distribution]]. Also known as the [[Vilfredo Pareto|Pareto]]-Zipf law, it is a [[power-law]] distribution on ranked data, named after the [[linguistics|linguist]] [[George Kingsley Zipf]] who suggested a simpler distribution called [[Zipf's law]], and the mathematician [[Benoît Mandelbrot]], who subsequently generalized it.
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| The [[probability mass function]] is given by:
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| :<math>f(k;N,q,s)=\frac{1/(k+q)^s}{H_{N,q,s}}</math>
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| where <math>H_{N,q,s}</math> is given by:
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| :<math>H_{N,q,s}=\sum_{i=1}^N \frac{1}{(i+q)^s}</math>
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| which may be thought of as a generalization of a [[harmonic number]]. In the formula, k is the rank of the data, and q and s are parameters of the distribution. In the limit as <math>N</math> approaches infinity, this becomes the [[Hurwitz zeta function]] <math>\zeta(s,q)</math>. For finite <math>N</math> and <math>q=0</math> the Zipf–Mandelbrot law becomes [[Zipf's law]]. For infinite <math>N</math> and <math>q=0</math> it becomes a [[Zeta distribution]].
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| ==Applications==
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| The distribution of words ranked by their [[frequency]] in a random
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| [[text corpus]] is generally a [[power-law]] distribution, known
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| as [[Zipf's law]].
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| If one plots the [[frequency]] rank of words contained in a large corpus of text data versus the number of occurrences or actual frequencies, one obtains a [[power-law]] distribution, with [[exponent]] close to one (but see Gelbukh & Sidorov, 2001).
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| In ecological field studies, the [[relative abundance distribution]] (i.e. the graph of the number of species observed as a function of their abundance) is often found to conform to a Zipf–Mandelbrot law.<ref>{{cite journal | last = Mouillot | first = D | coauthors = Lepretre, A
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| | title = Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity
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| | journal = Environmental Monitoring and Assessment | volume = 63 | issue = 2 | pages = 279–295
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| | publisher = Springer | year = 2000 | url = http://cat.inist.fr/?aModele=afficheN&cpsidt=1411186
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| | accessdate = 24 Dec 2008 | doi = 10.1023/A:1006297211561}}</ref>
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| Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandlebrot distributions.<ref>{{cite journal | last = Manaris | first = B | coauthors = Vaughan, D, Wagner, CS, Romero, J, Davis, RB | title = Evolutionary Music and the Zipf-Mandelbrot Law: Developing Fitness Functions for Pleasant Music | journal = Proceedings of 1st European Workshop on Evolutionary Music and Art (EvoMUSART2003) | volume = 611 | url = http://shaunwagner.com/writings_computer_evomus.html}}</ref>
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| ==Notes==
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| {{reflist}}
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| == References==
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| * {{Cite book
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| | last = Mandelbrot | first = Benoît | authorlink = Benoît Mandelbrot
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| | chapter = Information Theory and Psycholinguistics
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| | title = Scientific psychology
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| | editor= B.B. Wolman and E. Nagel
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| | year = 1965
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| | publisher = Basic Books
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| }} Reprinted as
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| ** {{Cite book
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| | last = Mandelbrot | first = Benoît | authorlink = Benoît Mandelbrot
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| | chapter = Information Theory and Psycholinguistics
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| | title = Language
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| | editor= R.C. Oldfield and J.C. Marchall
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| | year = 1968
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| | origyear = 1965
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| | publisher = Penguin Books
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| }}
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| * {{cite book
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| | last = Zipf
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| | first = George Kingsley
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| | authorlink = George Kingsley Zipf
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| | title = Selected Studies of the Principle of Relative Frequency in Language
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| | publisher = Harvard University Press
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| | year = 1932
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| | location = Cambridge, MA}}
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| ==External links==
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| * [http://arxiv.org/abs/physics/9901035 Z. K. Silagadze: Citations and the Zipf-Mandelbrot's law]
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| * [http://www.nist.gov/dads/HTML/zipfslaw.html NIST: Zipf's law]
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| * [http://www.nslij-genetics.org/wli/zipf/index.html W. Li's References on Zipf's law]
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| * [http://www.gelbukh.com/CV/Publications/2001/CICLing-2001-Zipf.htm Gelbukh & Sidorov, 2001: Zipf and Heaps Laws’ Coefficients Depend on Language]
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| {{ProbDistributions|discrete-finite}}
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| {{DEFAULTSORT:Zipf-Mandelbrot Law}}
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| [[Category:Discrete distributions]]
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| [[Category:Power laws]]
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| [[Category:Computational linguistics]]
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| [[Category:Quantitative linguistics]]
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| [[Category:Probability distributions]]
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