Logarithmic distribution

In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution.

The logarithmic distribution is derived from the Maclaurin series expansion of ln(1 − p), which is

$\ln(1-p) = -p - \frac{p^2}{2} - \frac{p^3}{3} - \cdots$

From this we obtain the identity

$\sum_{k=0}^{\infty} \frac{1}{-\ln(1-p)} \; \frac{p^k}{k} = 1$ .

This leads directly to the probability mass function of a Log(p)-distributied 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

$F(k) = 1 + \frac{\mathrm{B}_p(k+1,0)}{\ln(1-p)}$

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 X_i, 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.

Logarithmic distribution

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