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Logarithmic distribution

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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 + \frac + \cdots. ]
From this we obtain the identity

[\sum_^ \frac \; \frac = 1. ]
This leads directly to the probability mass function of a Log(p)-distributed random variable:

[ f(k) = \frac \; \frac]
for [k \ge 1], and where [0 The cumulative distribution function is

[ F(k) = 1 + \frac]
where [\Beta] 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 [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 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

 Probability distributions  [ view] • [ talk] • [ edit] 
Univariate Multivariate
Discrete: Bernoulli • binomial • Boltzmann • compound Poisson • degenerate • degree • 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 • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • 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 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

 

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