Logarithmic distribution

name =Logarithmic| type =mass| pdf_image =[[Image:Logarithmicpmf.svg|300px|center|Plot of the logarithmic PMF]]The function is only defined at integer values. The connecting lines are merely guides for the eye. | cdf_image =[[Image:Logarithmiccdf.svg|300px|center|Plot of the logarithmic CDF]]| parameters =0 | support =k \in {1,2,3,\ldots}| 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 =- \frac{p^2 + p\ln(1-p)}{(1-p)^2(\ln(1-p))^2}| skewness =| kurtosis =| entropy =| mgf =\frac{\ln(1 - pe^t)}{\ln(1-p)}\text{ for }t | char =\frac{\ln(1 - pe^{it})}{\ln(1-p)}| pgf =\frac{\ln(1-pz)}{\ln(1-p)}\text{ for }|z| |

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

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 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.{{Cite journal |url-status = dead |archive-url = https://web.archive.org/web/20110726144520/http://www.math.mcgill.ca/~dstephens/556/Papers/Fisher1943.pdf |archive-date = 2011-07-26

References