page_title: Yule–Simon distribution
hierarchical_section_title: Yule–Simon distribution
caption_attribution_description: English: en:gnuplot source available on request
context_page_description: 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.
  The probability mass function of the Yule–Simon distribution is
  for integer and real, where is the beta function. Equivalently the pmf can be written in terms of the rising factorial as
  where is the gamma function. Thus, if is an integer,
  The parameter can be estimated using a fixed point algorithm.
  The probability mass function f has the property that for sufficiently large k we have
  This means that the tail of the Yule–Simon distribution is a realization of Zipf's law: can be used to model, for example, the relative frequency of the th most frequent word in a large collection of text, which according to Zipf's law is inversely proportional to a power of.
context_section_description: 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.
  The probability mass function (pmf) of the Yule–Simon (ρ) distribution is
  for integer  and real , where  is the beta function.  Equivalently the pmf can be written in terms of the rising factorial as
  where  is the gamma function.  Thus, if  is an integer,
  The parameter  can be estimated using a fixed point algorithm.
  The probability mass function f has the property that for sufficiently large k we have
  This means that the tail of the Yule–Simon distribution is a realization of Zipf's law:  can be used to model, for example, the relative frequency of the 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 .
caption: Plot of the Yule–Simon(1) distribution (red) and its asymptotic Zipf's law (blue)
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