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

This leads directly to the probability mass function of a Log(p)-distributed random variable:

for k ≥ 1, and where 0 < p < 1. Because of the identity above, the distribution is properly normalized.

The cumulative distribution function is

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

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.

Famous quotes containing the word distribution:

“Classical and romantic: private language of a family quarrel, a dead dispute over the distribution of emphasis between man and nature.”
—Cyril Connolly (1903–1974)