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 + p 2 2 + p 3 3 + ⋯ . {\displaystyle -\ln(1-p)=p+{\frac {p^{2}}{2}}+{\frac {p^{3}}{3}}+\cdots .}
From this we obtain the identity
∑ k = 1 ∞ − 1 ln ( 1 − p ) p k k = 1. {\displaystyle \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 ) = − 1 ln ( 1 − p ) p k k {\displaystyle 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 + B ( p ; k + 1 , 0 ) ln ( 1 − p ) {\displaystyle F(k)=1+{\frac {\mathrm {B} (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
∑ i = 1 N X i {\displaystyle \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.
See also Poisson distribution (also derived from a Maclaurin series)
References
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.
