Logarithmic distribution: Definition from Answers.com

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	$0 < p < 1\!$
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 or the log-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 ≥ 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 B 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

$\sum_{n=1}^N X_i$

has a negative binomial distribution. In this way, the negative binomial distribution is seen to be a compound Poisson distribution.

R.A. Fisher described the logarithmic distribution in a paper that used it to model relative species abundance.^[1]

References

^ Fisher, R.A.; Corbet, A.S.; Williams, C.B. (1943), "The Relation Between the Number of Species and the Number of Individuals in a Random Sample of an Animal Population", Journal of Animal Ecology 12 (1): 42–58, JSTOR: 1411, http://www.math.mcgill.ca/~dstephens/556/Papers/Fisher1943.pdf

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (-∞,∞)

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · stable · Tweedie

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Logarithmic distribution

See also

References

Further reading

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