Inner model - Revision history

208.91.114.4: punctuation

2014-05-09T20:57:07Z

punctuation

← Older revision		Revision as of 22:57, 9 May 2014
Line 1:		Line 1:
	~~{{Probability distribution\|~~		std home [http://Www.aahealth.org/programs/comm-diseases/hiv-aids/hotlines test Nice] to meet you, my title is Figures Held although I don't truly like being known as like that. I am a meter reader std testing at home but I plan on changing it. One of the things she loves most is to study comics and she'll be starting something else alongside with it. Minnesota has usually been his house but his wife wants them to transfer.<br><br>Feel free to visit my home std test kit blog post - home [http://Www.Ranker.com/list/famous-people-with-herpes/celebrity-lists std test] kit [[http://www.dinnerfor6.com/home/profile/DLangridg Read Homepage]]
	~~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'' ≥ 1,~~ and ~~where 0 < ~~'~~'p'' < 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~~]]

en>Mark viking: Added wiki links, moved model wiki link to first occurence

2013-04-26T00:13:58Z

Added wiki links, moved model wiki link to first occurence

← Older revision		Revision as of 02:13, 26 April 2013
Line 1:		Line 1:
	~~Nice to satisfy you~~, ~~I am Marvella Shryock. For many years I~~'~~ve been working~~ as a ~~payroll clerk~~. ~~South Dakota is exactly~~ where I'~~ve usually been living~~. ~~One~~ of the ~~issues she loves most~~ is ~~to do aerobics and now she~~ is ~~trying to earn money~~ with it.<br><br>~~my homepage -~~ [http://~~Uin~~.cc/~~dietfooddelivery78594 uin~~.cc]		{{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'' ≥ 1, and where 0 < ''p'' < 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]]

en>LaaknorBot: Robot: Adding zh:內模型

2012-05-27T17:09:58Z

Robot: Adding zh:內模型

New page

Nice to satisfy you, I am Marvella Shryock. For many years I've been working as a payroll clerk. South Dakota is exactly where I've usually been living. One of the issues she loves most is to do aerobics and now she is trying to earn money with it.<br><br>my homepage - [http://Uin.cc/dietfooddelivery78594 uin.cc]

← Older revision		Revision as of 22:57, 9 May 2014
Line 1:		Line 1:
	~~{{Probability distribution\|~~		std home [http://Www.aahealth.org/programs/comm-diseases/hiv-aids/hotlines test Nice] to meet you, my title is Figures Held although I don't truly like being known as like that. I am a meter reader std testing at home but I plan on changing it. One of the things she loves most is to study comics and she'll be starting something else alongside with it. Minnesota has usually been his house but his wife wants them to transfer.<br><br>Feel free to visit my home std test kit blog post - home [http://Www.Ranker.com/list/famous-people-with-herpes/celebrity-lists std test] kit [[http://www.dinnerfor6.com/home/profile/DLangridg Read Homepage]]
	~~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'' ≥ 1,~~ and ~~where 0 < ~~'~~'p'' < 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~~]]