# Logarithmic distribution

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

$-\ln(1-p)=p+{\frac {p^{2}}{2}}+{\frac {p^{3}}{3}}+\cdots .$ From this we obtain the identity

$\sum _{k=1}^{\infty }{\frac {-1}{\ln(1-p)}}\;{\frac {p^{k}}{k}}=1.$ This leads directly to the probability mass function of a Log(p)-distributed random variable:

$f(k)={\frac {-1}{\ln(1-p)}}\;{\frac {p^{k}}{k}}$ for $k\geq 1$ , and where $0 . Because of the identity above, the distribution is properly normalized.

$F(k)=1+{\frac {\mathrm {B} _{p}(k+1,0)}{\ln(1-p)}}$ where $\mathrm {B}$ is the incomplete beta function.

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

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