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
: <math>
-\ln(1-p) = p + \frac{p^2}{2} + \frac{p^3}{3} + \cdots.
</math>
From this we obtain the identity
:<math>\sum_{k=1}^{\infty} \frac{-1}{\ln(1-p)} \; \frac{p^k}{k} = 1. </math>
This leads directly to the probability mass function of a Log(p)-distributed random variable:
:<math> f(k) = \frac{-1}{\ln(1-p)} \; \frac{p^k}{k}</math>
for k ≥ 1, and where 0 < p < 1. Because of the identity above, the distribution is properly normalized.
The cumulative distribution function is
:<math> F(k) = 1 + \frac{\Beta(p; k+1,0)}{\ln(1-p)}</math>
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<sub>i</sub>, i = 1, 2, 3, ... is an infinite sequence of independent identically distributed random variables each having a Log(p) distribution, then
:<math>\sum_{i=1}^N X_i</math>
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)