# Yule-Simon distribution

Parameters Probability mass functionPlot of the Yule-Simon PMFYule-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.) Cumulative distribution functionPlot of the Yule-Simon CMFYule-Simon CMF. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.) ${\displaystyle \rho >0\,}$ shape (real) ${\displaystyle k\in \{1,2,\dots \}\,}$ ${\displaystyle \rho \,\mathrm {B} (k,\rho +1)\,}$ ${\displaystyle 1-k\,\mathrm {B} (k,\rho +1)\,}$ ${\displaystyle {\frac {\rho }{\rho -1}}\,}$ for ${\displaystyle \rho >1\,}$ ${\displaystyle 1\,}$ ${\displaystyle {\frac {\rho ^{2}}{(\rho -1)^{2}\;(\rho -2)}}\,}$ for ${\displaystyle \rho >2\,}$ ${\displaystyle {\frac {(\rho +1)^{2}\;{\sqrt {\rho -2}}}{(\rho -3)\;\rho }}\,}$ for ${\displaystyle \rho >3\,}$ ${\displaystyle \rho +3+{\frac {11\rho ^{3}-49\rho -22}{(\rho -4)\;(\rho -3)\;\rho }}\,}$ for ${\displaystyle \rho >4\,}$ ${\displaystyle {\frac {\rho }{\rho +1}}\;{}_{2}F_{1}(1,1;\rho +2;e^{t})\,e^{t}\,}$ ${\displaystyle {\frac {\rho }{\rho +1}}\;{}_{2}F_{1}(1,1;\rho +2;e^{i\,t})\,e^{i\,t}\,}$

In probability and statistics, the Yule-Simon distribution is a discrete probability distribution named after Udny Yule and Herbert Simon. Simon originally called it the Yule distribution.

The probability mass function of the Yule-Simon(ρ) distribution is

${\displaystyle f(k;\rho )=\rho \,\mathrm {B} (k,\rho +1),\,}$

for integer ${\displaystyle k\geq 1}$ and real ${\displaystyle \rho >0}$, where ${\displaystyle \mathrm {B} }$ is the beta function. Equivalently the pmf can be written in terms of the falling factorial as

${\displaystyle f(k;\rho )={\frac {\rho \,\Gamma (\rho +1)}{(k+\rho )^{\underline {\rho +1}}}},\,}$

where ${\displaystyle \Gamma }$ is the gamma function. Thus, if ${\displaystyle \rho }$ is an integer,

${\displaystyle f(k;\rho )={\frac {\rho \,\rho !\,(k-1)!}{(k+\rho )!}}.\,}$

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

${\displaystyle f(k;\rho )\approx {\frac {\rho \,\Gamma (\rho +1)}{k^{\rho +1}}}\propto {\frac {1}{k^{\rho +1}}}.\,}$

This means that the tail of the Yule-Simon distribution is a realization of Zipf's law: ${\displaystyle f(k;\rho )}$ can be used to model, for example, the relative frequency of the ${\displaystyle k}$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 ${\displaystyle k}$.

## Occurrence

The Yule-Simon distribution arises as a continuous mixture of geometric distributions. Specifically, assume that ${\displaystyle W}$ follows an exponential distribution with scale ${\displaystyle 1/\rho }$ or rate ${\displaystyle \rho }$:

${\displaystyle W\sim \mathrm {Exponential} (\rho )\,}$
${\displaystyle h(w;\rho )=\rho \,\exp(-\rho \,w)\,}$

Then a Yule-Simon distributed variable ${\displaystyle K}$ has the following geometric distribution:

${\displaystyle K\sim \mathrm {Geometric} (\exp(-W))\,}$

The pmf of a geometric distribution is

${\displaystyle g(k;p)=p\,(1-p)^{k-1}\,}$

for ${\displaystyle k\in \{1,2,\dots \}}$. The Yule-Simon pmf is then the following exponential-geometric mixture distribution:

${\displaystyle f(k;\rho )=\int _{0}^{\infty }\,\,\,g(k;\exp(-w))\,h(w;\rho )\,dw\,}$

## Generalizations

The two-parameter generalization of the original Yule distribution replaces the beta function with an incomplete beta function. The probability mass function of the generalized Yule-Simon(ρ, α) distribution is defined as

${\displaystyle f(k;\rho ,\alpha )={\frac {\rho }{1-\alpha ^{\rho }}}\;\mathrm {B} _{1-\alpha }(k,\rho +1),\,}$

with ${\displaystyle 0\leq \alpha <1}$. For ${\displaystyle \alpha =0}$ the ordinary Yule-Simon(ρ) distribution is obtained as a special case. The use of the incomplete beta function has the effect of introducing an exponential cutoff in the upper tail.

File:Yule-Simon distribution.png
Plot of the Yule-Simon(1) distribution (red) and its asymptotic Zipf law (blue)

## References

• Herbert A. Simon, On a Class of Skew Distribution Functions, Biometrika 42(3/4): 425–440, December 1955.
• Colin Rose and Murray D. Smith, Mathematical Statistics with Mathematica. New York: Springer, 2002, ISBN 0-387-95234-9. (See page 107, where it is called the "Yule distribution".)