Logarithmic distribution

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 $$k \ge 1$$, and where $$0<p<1$$. Because of the identity above, the distribution is properly normalized.

The cumulative distribution function is


 * $$ F(k) = 1 + \frac{\Beta_p(k+1,0)}{\ln(1-p)}$$

where $$\Beta$$ 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.