Probability mass function
The function is only defined at integer values. The connecting lines are merely guides for the eye.
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.
The probability mass function ƒ of this distribution satisfies the recurrence relation
- 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" (PDF). Journal of Animal Ecology. 12 (1): 42–58. doi:10.2307/1411. JSTOR 1411.
- 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.