Logarithmic distribution

Discrete probability distribution

Logarithmic
Probability mass function Plot of the logarithmic PMF Plot of the logarithmic PMF The function is only defined at integer values. The connecting lines are merely guides for the eye.
Cumulative distribution function Plot of the logarithmic CDF Plot of the logarithmic CDF
Parameters	$0<p<1$ {\displaystyle 0<p<1}
Support	$k\in \{1,2,3,\ldots \}$ {\displaystyle k\in \{1,2,3,\ldots \}}
PMF	${\frac {-1}{\ln(1-p)}}{\frac {p^{k}}{k}}$ {\displaystyle {\frac {-1}{\ln(1-p)}}{\frac {p^{k}}{k}}}
CDF	$1+{\frac {\mathrm {B} (p;k+1,0)}{\ln(1-p)}}$ {\displaystyle 1+{\frac {\mathrm {B} (p;k+1,0)}{\ln(1-p)}}}
Mean	${\frac {-1}{\ln(1-p)}}{\frac {p}{1-p}}$ {\displaystyle {\frac {-1}{\ln(1-p)}}{\frac {p}{1-p}}}
Mode	$1$ {\displaystyle 1}
Variance	$-{\frac {p^{2}+p\ln(1-p)}{(1-p)^{2}(\ln(1-p))^{2}}}$ {\displaystyle -{\frac {p^{2}+p\ln(1-p)}{(1-p)^{2}(\ln(1-p))^{2}}}}
MGF	${\frac {\ln(1-pe^{t})}{\ln(1-p)}}{\text{ for }}t<-\ln p$ {\displaystyle {\frac {\ln(1-pe^{t})}{\ln(1-p)}}{\text{ for }}t<-\ln p}
CF	${\frac {\ln(1-pe^{it})}{\ln(1-p)}}$ {\displaystyle {\frac {\ln(1-pe^{it})}{\ln(1-p)}}}
PGF	${\frac {\ln(1-pz)}{\ln(1-p)}}{\text{ for }}\|z\|<{\frac {1}{p}}$ {\displaystyle {\frac {\ln(1-pz)}{\ln(1-p)}}{\text{ for }}\|z\|<{\frac {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 .

{\displaystyle -\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.

{\displaystyle \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}}

{\displaystyle 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)}}

{\displaystyle 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}

{\displaystyle \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]

References

[edit ]

^ 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. Archived from the original (PDF) on 2011年07月26日.

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
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Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized normal Generalized hyperbolic Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's S_U Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda