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
\sum_{k=1}^{\infty} \frac{-1}{\ln(1-p)} \; \frac{p^k}{k} = 1.
This leads directly to the probability mass function of a Log(p)-distributed random variable:
f(k) = \frac{-1}{\ln(1-p)} \; \frac{p^k}{k}
for k ≥ 1, and where 0 < p < 1. Because of the identity above, the distribution is properly normalized.
The cumulative distribution function is
F(k) = 1 + \frac{\Beta(p; k+1,0)}{\ln(1-p)}
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
\sum_{i=1}^N X_i
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.[1]
See also
- Poisson distribution (also derived from a Maclaurin series)
References
- ^ 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. Bibcode:1943JAnEc..12...42F. doi:10.2307/1411. JSTOR 1411. Archived from the original on 2011-07-26.
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.