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

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

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.

See also

• Poisson distribution (also derived from a Maclaurin series)