# Logarithmic distribution

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

Parameters Probability mass function The function is only defined at integer values. The connecting lines are merely guides for the eye. Cumulative distribution function $0 $k\in \{1,2,3,\ldots \}$ ${\frac {-1}{\ln(1-p)}}{\frac {p^{k}}{k}}$ $1+{\frac {\mathrm {B} (p;k+1,0)}{\ln(1-p)}}$ ${\frac {-1}{\ln(1-p)}}{\frac {p}{1-p}}$ $1$ $-{\frac {p^{2}+p\ln(1-p)}{(1-p)^{2}(\ln(1-p))^{2}}}$ ${\frac {\ln(1-pe^{t})}{\ln(1-p)}}{\text{ for }}t<-\ln p$ ${\frac {\ln(1-pe^{it})}{\ln(1-p)}}$ ${\frac {\ln(1-pz)}{\ln(1-p)}}{\text{ for }}|z|<{\frac {1}{p}}$ $-\ln(1-p)=p+{\frac {p^{2}}{2}}+{\frac {p^{3}}{3}}+\cdots .$ 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 ≥ 1, and where 0 < p < 1. Because of the identity above, the distribution is properly normalized.

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

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