Logarithmic distribution

Logarithmic

Parameters0 < p < 1 {\displaystyle 0<p<1}

Supportk ∈ { 1 , 2 , 3 , … } {\displaystyle k\in \{1,2,3,\ldots \}}

PMF− 1 ln ⁡ ( 1 − p ) p k k {\displaystyle {\frac {-1}{\ln(1-p)}}{\frac {p^{k}}{k}}}

CDF1 + B ( p ; k + 1 , 0 ) ln ⁡ ( 1 − p ) {\displaystyle 1+{\frac {\mathrm {B} (p;k+1,0)}{\ln(1-p)}}}

Mean− 1 ln ⁡ ( 1 − p ) p 1 − p {\displaystyle {\frac {-1}{\ln(1-p)}}{\frac {p}{1-p}}}

Mode1 {\displaystyle 1}

Variance− p 2 + p ln ⁡ ( 1 − p ) ( 1 − p ) 2 ( ln ⁡ ( 1 − p ) ) 2 {\displaystyle -{\frac {p^{2}+p\ln(1-p)}{(1-p)^{2}(\ln(1-p))^{2}}}}

MGFln ⁡ ( 1 − p e t ) ln ⁡ ( 1 − p ) for t < − ln ⁡ p {\displaystyle {\frac {\ln(1-pe^{t})}{\ln(1-p)}}{\text{ for }}t<-\ln p}

CFln ⁡ ( 1 − p e i t ) ln ⁡ ( 1 − p ) {\displaystyle {\frac {\ln(1-pe^{it})}{\ln(1-p)}}}

PGFln ⁡ ( 1 − p z ) ln ⁡ ( 1 − p ) for | z | < 1 p {\displaystyle {\frac {\ln(1-pz)}{\ln(1-p)}}{\text{ for }}|z|<{\frac {1}{p}}}

In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion

− ln ⁡ ( 1 − p ) = p + p 2 2 + p 3 3 + ⋯ . {\displaystyle -\ln(1-p)=p+{\frac {p^{2}}{2}}+{\frac {p^{3}}{3}}+\cdots .}

From this we obtain the identity

∑ k = 1 ∞ − 1 ln ⁡ ( 1 − p ) p k k = 1. {\displaystyle \sum _{k=1}^{\infty }{\frac {-1}{\ln(1-p)}}\;{\frac {p^{k}}{k}}=1.}

This leads directly to the probability mass function of a Log(p)-distributed random variable:

f ( k ) = − 1 ln ⁡ ( 1 − p ) p k k {\displaystyle f(k)={\frac {-1}{\ln(1-p)}}\;{\frac {p^{k}}{k}}}

for k ≥ 1, and where 0 < p < 1. Because of the identity above, the distribution is properly normalized.

The cumulative distribution function is

F ( k ) = 1 + B ( p ; k + 1 , 0 ) ln ⁡ ( 1 − p ) {\displaystyle F(k)=1+{\frac {\mathrm {B} (p;k+1,0)}{\ln(1-p)}}}

where B is the incomplete beta function.

A Poisson compounded with Log(p)-distributed random variables has a negative binomial distribution. In other words, if N is a random variable with a Poisson distribution, and Xi, i = 1, 2, 3, ... is an infinite sequence of independent identically distributed random variables each having a Log(p) distribution, then

∑ i = 1 N X i {\displaystyle \sum _{i=1}^{N}X_{i}}

has a negative binomial distribution. In this way, the negative binomial distribution is seen to be a compound Poisson distribution.

R. A. Fisher described the logarithmic distribution in a paper that used it to model relative species abundance.

Logarithmic distribution

See also

Further reading