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| {{Probability distribution|
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| name =Logarithmic|
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| type =mass|
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| 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> |
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| cdf_image =[[Image:Logarithmiccdf.svg|300px|center|Plot of the logarithmic CDF]]|
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| parameters =<math>0 < p < 1\!</math>|
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| support =<math>k \in \{1,2,3,\dots\}\!</math>|
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| pdf =<math>\frac{-1}{\ln(1-p)} \; \frac{\;p^k}{k}\!</math>|
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| cdf =<math>1 + \frac{\Beta(p;k+1,0)}{\ln(1-p)}\!</math>|
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| mean =<math>\frac{-1}{\ln(1-p)} \; \frac{p}{1-p}\!</math>|
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| median =|
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| mode =<math>1</math>|
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| variance =<math>-p \;\frac{p + \ln(1-p)}{(1-p)^2\,\ln^2(1-p)} \!</math>|
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| skewness =<!-- exists, but too complex -->|
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| kurtosis =<!-- exists, but too complex -->|
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| entropy =<!-- exists, but too complex -->|
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| mgf =<math>\frac{\ln(1 - p\,\exp(t))}{\ln(1-p)}\text{ for }t<-\ln p\,</math>|
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| char =<math>\frac{\ln(1 - p\,\exp(i\,t))}{\ln(1-p)}\text{ for }t\in\mathbb{R}\!</math>|
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| pgf =<math>\frac{\ln(1-pz)}{\ln(1-p)}\text{ for }|z|<\frac1p</math>|
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| }}
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| 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
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| : <math>
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| -\ln(1-p) = p + \frac{p^2}{2} + \frac{p^3}{3} + \cdots. | |
| </math>
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| From this we obtain the identity
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| :<math>\sum_{k=1}^{\infty} \frac{-1}{\ln(1-p)} \; \frac{p^k}{k} = 1. </math>
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| This leads directly to the [[probability mass function]] of a Log(''p'')-distributed [[random variable]]:
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| :<math> f(k) = \frac{-1}{\ln(1-p)} \; \frac{p^k}{k}</math>
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| for ''k'' ≥ 1, and where 0 < ''p'' < 1. Because of the identity above, the distribution is properly normalized.
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| The [[cumulative distribution function]] is
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| :<math> F(k) = 1 + \frac{\Beta(p; k+1,0)}{\ln(1-p)}</math>
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| where ''B'' is the [[incomplete beta function]].
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| 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
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| :<math>\sum_{i=1}^N X_i</math>
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| has a negative binomial distribution. In this way, the negative binomial distribution is seen to be a [[compound Poisson distribution]].
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| [[Ronald Fisher|R.A. Fisher]] described the logarithmic distribution in a paper that used it to model [[relative species abundance]].<ref>{{Cite journal
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| | doi = 10.2307/1411
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| | title = The Relation Between the Number of Species and the Number of Individuals in a Random Sample of an Animal Population
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| | jstor = 1411
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| | url = http://www.math.mcgill.ca/~dstephens/556/Papers/Fisher1943.pdf
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| | year = 1943
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| | journal = Journal of Animal Ecology
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| | pages = 42–58
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| | volume = 12
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| | issue = 1
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| | last1 = Fisher | first1 = R.A.
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| | last2 = Corbet | first2 = A.S.
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| | last3 = Williams | first3 = C.B.
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| | postscript = <!--None-->
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| }}</ref>
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| ==See also==
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| * [[Poisson distribution]] (also derived from a Maclaurin series)
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| ==References==
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| <references/>
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| ==Further reading==
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| *{{cite book|last=Johnson|first=Norman Lloyd|coauthors=Kemp, Adrienne W; Kotz, Samuel|title=Univariate discrete distributions|publisher=John Wiley & Sons|year=2005|edition=3|chapter=Chapter 7: Logarithmic and Lagrangian distributions|isbn=978-0-471-27246-5}}
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| *{{MathWorld|urlname=Log-SeriesDistribution|title=Log-Series Distribution}}
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| {{ProbDistributions|discrete-infinite}}
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| [[Category:Discrete distributions]]
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| [[Category:Logarithms]] | |
| [[Category:Probability distributions]]
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