Logarithmic distribution

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Logarithmic
Probability mass function
Cumulative distribution function
Parameters 0 < p < 1\!
Support k \in \{1,2,3,\dots\}\!
Probability mass function (pmf) \frac{-1}{\ln(1-p)} \; \frac{\;p^k}{k}\!
Cumulative distribution function (cdf) 1 + \frac{\Beta_p(k+1,0)}{\ln(1-p)}\!
Mean \frac{-1}{\ln(1-p)} \; \frac{p}{1-p}\!
Median
Mode 1
Variance -p \;\frac{p + \ln(1-p)}{(1-p)^2\,\ln^2(1-p)} \!
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf) \frac{\ln(1 - p\,\exp(t))}{\ln(1-p)}\!
Characteristic function \frac{\ln(1 - p\,\exp(i\,t))}{\ln(1-p)}\!

In probability and statistics, the logarithmic distribution (also known as the logarithmic 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 \ge 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 Β is the incomplete beta function.

A Poisson mixture of 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_{n=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 applied this distribution to population genetics.

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