Logarithmic distribution

Short description: Discrete probability distribution

Logarithmic

Probability mass function The function is only defined at integer values. The connecting lines are merely guides for the eye.
Cumulative distribution function
Parameters	[math]\displaystyle{ 0 \lt p \lt 1 }[/math]
Support	[math]\displaystyle{ k \in \{1,2,3,\ldots\} }[/math]
pmf	[math]\displaystyle{ \frac{-1}{\ln(1-p)} \frac{p^k}{k} }[/math]
CDF	[math]\displaystyle{ 1 + \frac{\Beta(p;k+1,0)}{\ln(1-p)} }[/math]
Mean	[math]\displaystyle{ \frac{-1}{\ln(1-p)} \frac{p}{1-p} }[/math]
Mode	[math]\displaystyle{ 1 }[/math]
Variance	[math]\displaystyle{ - \frac{p^2 + p\ln(1-p)}{(1-p)^2(\ln(1-p))^2} }[/math]
MGF	[math]\displaystyle{ \frac{\ln(1 - pe^t)}{\ln(1-p)}\text{ for }t \lt -\ln p }[/math]
CF	[math]\displaystyle{ \frac{\ln(1 - pe^{it})}{\ln(1-p)} }[/math]
PGF	[math]\displaystyle{ \frac{\ln(1-pz)}{\ln(1-p)}\text{ for }|z| \lt \frac{1}{p} }[/math]

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

[math]\displaystyle{ -\ln(1-p) = p + \frac{p^2}{2} + \frac{p^3}{3} + \cdots. }[/math]

From this we obtain the identity

[math]\displaystyle{ \sum_{k=1}^{\infty} \frac{-1}{\ln(1-p)} \; \frac{p^k}{k} = 1. }[/math]

This leads directly to the probability mass function of a Log(p)-distributed random variable:

[math]\displaystyle{ f(k) = \frac{-1}{\ln(1-p)} \; \frac{p^k}{k} }[/math]

for k ≥ 1, and where 0 < p < 1. Because of the identity above, the distribution is properly normalized.

The cumulative distribution function is

[math]\displaystyle{ F(k) = 1 + \frac{\Beta(p; k+1,0)}{\ln(1-p)} }[/math]

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 X_i, i = 1, 2, 3, ... is an infinite sequence of independent identically distributed random variables each having a Log(p) distribution, then

[math]\displaystyle{ \sum_{i=1}^N X_i }[/math]

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)

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". http://mathworld.wolfram.com/Log-SeriesDistribution.html.