|Probability mass function
|Cumulative distribution function
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
From this we obtain the identity
for k ≥ 1, and where 0 < p < 1. Because of the identity above, the distribution is properly normalized.
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 Xi, 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.
See also 
- Poisson distribution (also derived from a Maclaurin series)
- Fisher, R.A.; Corbet, A.S.; Williams, C.B. (1943). "The Relation Between the Number of Species and the Number of Individuals in a Random Sample of an Animal Population". Journal of Animal Ecology 12 (1): 42–58. doi:10.2307/1411. JSTOR 1411
Further reading 
- Johnson, Norman Lloyd; Kemp, Adrienne W; Kotz, Samuel (2005). "Chapter 7: Logarithmic and Lagrangian distributions". Univariate discrete distributions (3 ed.). John Wiley & Sons. ISBN 978-0-471-27246-5.
- Weisstein, Eric W., "Log-Series Distribution", MathWorld.