Logarithmic 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	$0<p<1\!$
Support	$k\in \{1,2,3,\dots \}\!$
pmf	${\frac {-1}{\ln(1-p)}}\;{\frac {\;p^{k}}{k}}\!$
CDF	$1+{\frac {\mathrm{B} (p;k+1,0)}{\ln(1-p)}}\!$
Mean	${\frac {-1}{\ln(1-p)}}\;{\frac {p}{1-p}}\!$
Mode	$1$
Variance	$-p\;{\frac {p+\ln(1-p)}{(1-p)^{2}\,\ln ^{2}(1-p)}}\!$
MGF	${\frac {\ln(1-p\,\exp(t))}{\ln(1-p)}}{\text{ for }}t<-\ln p\,$
CF	${\frac {\ln(1-p\,\exp(i\,t))}{\ln(1-p)}}{\text{ for }}t\in {\mathbb {R}}\!$
PGF	${\frac {\ln(1-pz)}{\ln(1-p)}}{\text{ for }}\|z\|<{\frac 1p}$

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 {\mathrm{B} (p;k+1,0)}{\ln(1-p)}}

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

\sum _{{i=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]

The probability mass function ƒ of this distribution satisfies the recurrence relation

f(k+1)={\frac {kp}{k+1}}f(k);{\text{ with the initial value }}f(1)={\frac {-p}{\ln(1-p)}}.

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" (PDF). Journal of Animal Ecology. 12 (1): 42–58. doi:10.2307/1411. JSTOR 1411.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Logarithmic distribution

See also

References

Further reading