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
mgf \frac{\ln(1 - p\,\exp(t))}{\ln(1-p)}\!
Char. func. \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.

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=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.

[edit] See also

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Univariate Multivariate
Discrete: Bernoulli • binomial • Boltzmann • compound Poisson • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot Ewens • multinomial
Continuous: Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square • inverse Gaussian • inverse gamma • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda Dirichlet • Kent • matrix normal • multivariate normal • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling • singular
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