Yule–Simon distribution について

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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
:

f(k;\rho) = \rho\operatorname(k, \rho+1)

k \geq 1

and real

\rho > 0

, where

\operatorname

is the beta function. Equivalently the pmf can be written in terms of the falling factorial as
:

f(k;\rho) = \frac

,
where

\Gamma

is the gamma function. Thus, if

\rho

is an integer,
:

f(k;\rho) = \frac

.
The parameter

\rho

can be estimated using a fixed point algorithm.
The probability mass function ''f'' has the property that for sufficiently large ''k'' we have
:

f(k;\rho) \approx \frac{k^{\rho+1}}

.
This means that the tail of the Yule–Simon distribution is a realization of Zipf's law:

f(k;\rho)

can be used to model, for example, the relative frequency of the

k

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

k

.
==Occurrence==

The Yule–Simon distribution arose originally as the limiting distribution of a particular stochastic process studied by Yule as a model for the distribution of biological taxa and subtaxa.〔

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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