Editing Logarithmic distribution (section)

{{Short description|Discrete probability distribution}}
{{Probability distribution|
  name       =Logarithmic|
  type       =mass|
  pdf_image  =[[Image:Logarithmicpmf.svg|300px|center|Plot of the logarithmic PMF]]<small>The function is only defined at integer values. The connecting lines are merely guides for the eye.</small> |
  cdf_image  =[[Image:Logarithmiccdf.svg|300px|center|Plot of the logarithmic CDF]]|
  parameters =<math>0 < p < 1</math>|
  support    =<math>k \in \{1,2,3,\ldots\}</math>|
  pdf        =<math>\frac{-1}{\ln(1-p)} \frac{p^k}{k}</math>|
  cdf        =<math>1 + \frac{\Beta(p;k+1,0)}{\ln(1-p)}</math>|
  mean       =<math>\frac{-1}{\ln(1-p)} \frac{p}{1-p}</math>|
  median     =|
  mode       =<math>1</math>|
  variance   =<math>- \frac{p^2 + p\ln(1-p)}{(1-p)^2(\ln(1-p))^2}</math>|
  skewness   =<!-- exists, but too complex -->|
  kurtosis   =<!-- exists, but too complex -->|
  entropy    =<!-- exists, but too complex -->|
  mgf        =<math>\frac{\ln(1 - pe^t)}{\ln(1-p)}\text{ for }t < -\ln p</math>|
  char       =<math>\frac{\ln(1 - pe^{it})}{\ln(1-p)}</math>|
  pgf        =<math>\frac{\ln(1-pz)}{\ln(1-p)}\text{ for }|z| < \frac{1}{p}</math>|
}}

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

: <math>
 -\ln(1-p)  = p + \frac{p^2}{2} + \frac{p^3}{3} + \cdots.
</math>

From this we obtain the identity

:<math>\sum_{k=1}^{\infty} \frac{-1}{\ln(1-p)} \; \frac{p^k}{k} = 1. </math>

This leads directly to the [[probability mass function]] of a Log(''p'')-distributed [[random variable]]:

:<math> f(k) = \frac{-1}{\ln(1-p)} \; \frac{p^k}{k}</math>

for ''k''&nbsp;≥&nbsp;1, and where 0&nbsp;<&nbsp;''p''&nbsp;<&nbsp;1.  Because of the identity above, the distribution is properly normalized.

The [[cumulative distribution function]] is

:<math> F(k) = 1 + \frac{\Beta(p; k+1,0)}{\ln(1-p)}</math>

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''<sub>''i''</sub>, ''i'' = 1, 2, 3, ... is an infinite sequence of independent identically distributed random variables each having a Log(''p'') distribution, then

:<math>\sum_{i=1}^N X_i</math>
has a negative binomial distribution.  In this way, the negative binomial distribution is seen to be a [[compound Poisson distribution]].

[[Ronald Fisher|R. A. Fisher]] described the logarithmic distribution in a paper that used it to model [[relative species abundance]].<ref>{{Cite journal
 |doi         = 10.2307/1411
 |title       = The Relation Between the Number of Species and the Number of Individuals in a Random Sample of an Animal Population
 |jstor       = 1411
 |url         = http://www.math.mcgill.ca/~dstephens/556/Papers/Fisher1943.pdf
 |year        = 1943
 |journal     = Journal of Animal Ecology
 |pages       = 42–58
 |volume      = 12
 |issue       = 1
 |last1       = Fisher
 |first1      = R. A.
 |last2       = Corbet
 |first2      = A. S.
 |last3       = Williams
 |first3      = C. B.
 |bibcode = 1943JAnEc..12...42F
 |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
}}</ref>