Yule–Simon distribution

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

;

ρ

)

=

ρ

B

⁡

(

k

,

ρ

+

1

)

,

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

for integer

k

≥

1

{\displaystyle k\geq 1}

and real

ρ

>

0

{\displaystyle \rho >0}

, where

B

{\displaystyle \operatorname {B} }

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

f

(

k

;

ρ

)

=

ρ

Γ

(

ρ

+

1

)

(

k

+

ρ

)

ρ

+

1

_

,

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

where

Γ

{\displaystyle \Gamma }

is the gamma function. Thus, if

ρ

{\displaystyle \rho }

is an integer,

f

(

k

;

ρ

)

=

ρ

ρ

!

(

k

−

1

)

!

(

k

+

ρ

)

!

.

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

The parameter

ρ

{\displaystyle \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

;

ρ

)

≈

ρ

Γ

(

ρ

+

1

)

k

ρ

+

1

∝

1

k

ρ

+

1

.