# Logarithmic distribution

Parameters Probability mass function Cumulative distribution function ${\displaystyle 0 ${\displaystyle k\in \{1,2,3,\dots \}\!}$ ${\displaystyle {\frac {-1}{\ln(1-p)}}\;{\frac {\;p^{k}}{k}}\!}$ ${\displaystyle 1+{\frac {\mathrm {B} _{p}(k+1,0)}{\ln(1-p)}}\!}$ ${\displaystyle {\frac {-1}{\ln(1-p)}}\;{\frac {p}{1-p}}\!}$ ${\displaystyle 1}$ ${\displaystyle -p\;{\frac {p+\ln(1-p)}{(1-p)^{2}\,\ln ^{2}(1-p)}}\!}$ ${\displaystyle {\frac {\ln(1-p\,\exp(t))}{\ln(1-p)}}\!}$ ${\displaystyle {\frac {\ln(1-p\,\exp(i\,t))}{\ln(1-p)}}\!}$

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

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

From this we obtain the identity

${\displaystyle \sum _{k=1}^{\infty }{\frac {-1}{\ln(1-p)}}\;{\frac {p^{k}}{k}}=1.}$

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

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

for ${\displaystyle k\geq 1}$, and where ${\displaystyle 0. Because of the identity above, the distribution is properly normalized.

${\displaystyle F(k)=1+{\frac {\mathrm {B} _{p}(k+1,0)}{\ln(1-p)}}}$

where ${\displaystyle \mathrm {B} }$ is the incomplete beta function.

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

${\displaystyle \sum _{n=1}^{N}X_{i}}$

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

R.A. Fisher applied this distribution to population genetics.