Logarithmic distribution
# 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

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

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.

The cumulative distribution function is

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

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 $X_i,$ 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.

