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+
| p2 | |
| 2 |
+
| p3 | |
| 3 |
+ … .
From this we obtain the identity
| infty | |
| \sum | |
| k=1 |
| -1 | |
| ln(1-p) |
| pk | |
| k |
=1.
This leads directly to the probability mass function of a Log(p)-distributed random variable:
f(k)=
| -1 | |
| ln(1-p) |
| pk | |
| 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+
| \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
| N | |
| \sum | |
| i=1 |
Xi
R. A. Fisher described the logarithmic distribution in a paper that used it to model relative species abundance.[1]