Inner model: Difference between revisions

Revision as of 01:13, 26 April 2013

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} + \dots .

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 < p < 1. Because of the identity above, the distribution is properly normalized.

The cumulative distribution function is

F (k) = 1 + \frac{B (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.^[1]

References

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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

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[1]

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+{{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,\dots\}\!</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>-p \;\frac{p + \ln(1-p)}{(1-p)^2\,\ln^2(1-p)} \!</math>|
+  skewness   =<!-- exists, but too complex -->|
+  kurtosis   =<!-- exists, but too complex -->|
+  entropy    =<!-- exists, but too complex -->|
+  mgf        =<math>\frac{\ln(1 - p\,\exp(t))}{\ln(1-p)}\text{ for }t<-\ln p\,</math>|
+  char       =<math>\frac{\ln(1 - p\,\exp(i\,t))}{\ln(1-p)}\text{ for }t\in\mathbb{R}\!</math>|
+  pgf =<math>\frac{\ln(1-pz)}{\ln(1-p)}\text{ for }|z|<\frac1p</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.
+ | postscript =  <!--None-->
+ }}</ref>
+==See also==
+* [[Poisson distribution]] (also derived from a Maclaurin series)
+==References==
+<references/>
+==Further reading==
+*{{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}}
+*{{MathWorld|urlname=Log-SeriesDistribution|title=Log-Series Distribution}}
+{{ProbDistributions|discrete-infinite}}
+[[Category:Discrete distributions]]
+[[Category:Logarithms]]
+[[Category:Probability distributions]]

Inner model: Difference between revisions

Revision as of 01:13, 26 April 2013

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