Logarithmic distribution

Probability distribution
name =Logarithmic
type =mass
pdf_

cdf_

parameters =
support =$k\; in\; \{1,2,3,dots\}!$
pdf =$frac\{-1\}\{ln(1-p)\}\; ;\; frac\{;p^k\}\{k\}!$
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 =
kurtosis =
entropy =
mgf =$frac\{ln(1\; -\; p,exp(t))\}\{ln(1-p)\}!$
char =$frac\{ln(1\; -\; p,exp(i,t))\}\{ln(1-p)\}!$

In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution 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 $k\; ge\; 1$, and where

The cumulative distribution function is

:$F(k)\; =\; 1\; +\; frac\{Beta\_p(k+1,0)\}\{ln(1-p)\}$

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

:$sum\_\{n=1\}^N\; X\_i$

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.

ee also

* Poisson distribution (also derived from a Maclaurin series)