Logarithmic distribution
In probability and statistics, the logarithmic distribution is a discrete probability distribution derived from the Maclaurin series expansion
From this we obtain the identity
This leads directly to the probability mass function of a Log-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-distributed random variables has a negative binomial distribution. In other words, if N is a random variable with a Poisson distribution, and Xi, i = 1, 2, 3,... is an infinite sequence of independent identically distributed random variables each having a Log 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.
