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.

See also

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Probability distributions
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Discrete univariate with infinite support
Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta
de:Logarithmische Verteilung

it:Variabile casuale logaritmica


Acknowledgement and Attribution Regarding Sources of Content

Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .

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