Logarithmic distribution

Logarithmic distribution

Probability distribution
name =Logarithmic
type =mass
pdf_

cdf_

parameters =0 < p < 1!
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 0. 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 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)


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