Editing Yule–Simon distribution (section)

{{Short description|Discrete probability distribution}}
{{Probability distribution |
  name       =Yule–Simon|
  type       =mass|
  pdf_image  =[[File:Yule-Simon distribution PMF.svg|325px|Plot of the Yule–Simon PMF]]<br /><small>Yule–Simon 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  =[[File:Yule-Simon distribution CMF.svg|325px|Plot of the Yule–Simon CMF]]<br /><small>Yule–Simon CMF. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)</small>|
  parameters =<math>\rho>0\,</math> shape ([[real number|real]])|
  support    =<math>k \in \{1,2,\dotsc\}</math>|
  pdf        =<math>\rho\operatorname{B}(k, \rho+1)</math>|
  cdf        =<math>1 - k\operatorname{B}(k, \rho+1)</math>|
  mean       =<math>\frac \rho {\rho-1}</math> for <math>\rho>1</math>|
  median     =|
  mode       =<math>1</math>|
  variance   =<math>\frac{\rho^2}{(\rho-1)^2(\rho-2)}</math> for <math>\rho>2</math>|
  skewness   =<math>\frac{(\rho+1)^2\sqrt{\rho-2}}{(\rho-3)\rho}\,</math> for <math>\rho>3</math>|
  kurtosis   =<math>\rho+3+\frac{11\rho^3-49\rho-22} {(\rho-4)(\rho-3)\rho}</math> for <math>\rho>4</math>|
  entropy    =|
  mgf        = does not exist|
  char       =<math>\frac{\rho}{\rho+1}{}_2F_1(1,1; \rho+2; e^{i\,t})e^{i\,t}</math>|
}}
In [[probability]] and [[statistics]], the '''Yule–Simon distribution''' is a [[discrete probability distribution]] named after [[Udny Yule]] and [[Herbert A. Simon]].  Simon originally called it the '''''Yule distribution'''''.<ref name=SimonBiomet>{{cite journal
  | last = Simon
  | first = H. A.
  | title = On a class of skew distribution functions
  | journal = Biometrika
  | volume = 42
  | pages = 425&ndash;440
  | year = 1955
  | doi = 10.1093/biomet/42.3-4.425
  | issue = 3–4
}}</ref>

The [[probability mass function]] (pmf) of the Yule–Simon (''&rho;'') distribution is

:<math>f(k;\rho) = \rho\operatorname{B}(k, \rho+1),</math>

for [[integer]] <math>k \geq 1</math> and [[real number|real]] <math>\rho > 0</math>, where <math>\operatorname{B}</math> is the [[beta function]].  Equivalently the pmf can be written in terms of the [[Pochhammer symbol|rising factorial]] as

:<math>
 f(k;\rho) = \frac{\rho\Gamma(\rho+1)}{(k+\rho)^{\underline{\rho+1}}},
</math>

where <math>\Gamma</math> is the [[gamma function]].  Thus, if <math>\rho</math> is an integer,

:<math>
 f(k;\rho) = \frac{\rho\,\rho!\,(k-1)!}{(k+\rho)!}.
</math>

The parameter <math>\rho</math> can be estimated using a fixed point algorithm.<ref name=JMGGarcia>{{cite journal
| last = Garcia Garcia
| first = Juan Manuel
| title = A fixed-point algorithm to estimate the Yule-Simon distribution parameter
| journal = Applied Mathematics and Computation
| volume = 217
| issue = 21
| pages = 8560–8566
| year = 2011
| doi = 10.1016/j.amc.2011.03.092
| url = https://zenodo.org/record/848773
}}</ref>

The probability mass function ''f'' has the property that for sufficiently large ''k'' we have

:<math>
 f(k;\rho)
 \approx \frac{\rho\Gamma(\rho+1)}{k^{\rho+1}}
 \propto \frac 1 {k^{\rho+1}}.
</math>

[[File:Yule-Simon distribution.png|thumb|300px|Plot of the Yule–Simon(1) distribution (red) and its asymptotic Zipf's law (blue)]]
This means that the tail of the Yule–Simon distribution is a realization of [[Zipf's law]]: <math>f(k;\rho)</math> can be used to model, for example, the relative frequency of the <math>k</math>th most frequent word in a large collection of text, which according to Zipf's law is [[inversely proportional]] to a (typically small) power of <math>k</math>.