Inverted index - Revision history

en>Jaimemontoya at 18:35, 27 September 2014

2014-09-27T18:35:53Z

← Older revision		Revision as of 20:35, 27 September 2014
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	~~{{Probability distribution \|~~		Hello and welcome. My name is Figures Wunder. He used to be unemployed but now he is a meter reader. Body building is what my family and I enjoy. South Dakota is where I've usually been residing.<br><br>Also visit my blog :: [http://www.neweracinema.com/tube/user/KOPR http://www.neweracinema.com/tube/user/KOPR]
	~~name =Beta Prime\|~~
	~~type =density\|~~
	~~pdf_image =[[Image:Beta prime pdf~~.~~svg\|325px]]\|~~
	~~cdf_image =[[Image:Beta prime cdf.svg\|325px]]\|~~
	~~parameters =<math>\alpha > 0</math> [[shape parameter\|shape]] ([[real number\|real]])<br /><math>\beta > 0</math> shape (real)\|~~
	~~support =<math>x > 0\!</math>\|~~
	~~pdf =<math>f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}\!</math>\|~~
	~~cdf =<math> I_{\frac{x}{1+x}(\alpha,\beta) }</math> where <math>I_x(\alpha,\beta)</math> is the incomplete beta function\|~~
	~~mean =<math>\frac{\alpha}{\beta-1} \text{ if } \beta>1</math>\|~~
	~~median =\|~~
	~~mode =<math>\frac{\alpha-1}{\beta+1} \text{ if } \alpha\ge 1\text{, 0 otherwise}\!</math>\|~~
	~~variance =<math>\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2} \text{ if } \beta>2</math>\|~~
	~~skewness =<math>\frac{2(2\alpha+\beta-1)}{\beta-3}\sqrt{\frac{\beta-2}{\alpha(\alpha+\beta-1)}} \text{ if } \beta>3</math>\|~~
	~~kurtosis =\|~~
	~~entropy =\|~~
	~~mgf =\|~~
	~~char =\|~~
	}}

	~~In [[probability theory]] and [[statistics]], the '''beta prime distribution''' (also known as '''inverted beta distribution''' or '''beta distribution of the second kind'''<ref~~ name="Johnson et al 1995, p248">Johnson et al (1995), p248</ref>) is an [[probability distribution#Continuous probability distribution\|absolutely continuous probability distribution]] defined for <math>x > 0</math> with two parameters α and β, having the [[probability density function]]:

	~~: <math>f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}</math>~~

	~~where ''B''~~ is ~~a [[Beta function]]~~. ~~While the related [[beta distribution]]~~ is the [[conjugate prior distribution]] of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in [[odds]]. The distribution is a [[Pearson distribution#The Pearson type VI distribution\|Pearson type VI]] distribution.<ref name="Johnson et al 1995, p248"/>

	~~The mode of~~ a ~~variate ''X'' distributed as <math>\beta^{'}(\alpha,\beta)</math> is <math>\hat{X} = \frac{\alpha-1}{\beta+1}</math>~~.
	~~Its mean is <math>\frac{\alpha}{\beta-1}</math> if <math>\beta>1</math> (if <math>\beta \leq 1</math> the mean~~ is ~~infinite, in other words it has no well defined mean)~~
	and ~~its variance is~~
	~~<math>\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2}</math> if <math>\beta>2</math>~~.

	~~For <math>-\alpha <k <\beta </math>, the k-th moment <math> E[X^k] </math>~~ is ~~given by~~

	~~:<math> E[X^k]=\frac{B(\alpha+k,\beta-k)}{B(\alpha,\beta)}. </math>~~

	~~For <math> k\in \mathbb{N} </math> with <math>k <\beta </math>, this simplifies to~~

	~~:<math> E[X^k]=\prod_{i=1}^{k} \frac{\alpha+i-1}{\beta-i}. </math>~~

	~~The cdf can also be written as~~

	~~:<math> \frac{x^\alpha \cdot _2F_1(\alpha, \alpha+\beta, \alpha+1, -x)}{\alpha \cdot B(\alpha,\beta)}\!</math>~~
	where ~~<math>_2F_1</math> is the Gauss's hypergeometric function <sub>2</sub>F<sub>1</sub> .~~

	~~==Generalization==~~

	~~Two more parameters can be added to form the '''generalized beta prime distribution''~~'.

	:<~~math~~>~~p > 0</math> [[shape parameter\|shape]] ([[real number\|real]])~~ <br> ~~<math>q > 0</math> scale ([[real number\|real]])~~

	~~having the [[probability density function]]~~:

	: ~~<math>f(x;\alpha,\beta,p,q) = \frac{p{\left({\frac{x}{q}}\right)}^{\alpha p-1} \left({1+{\left({\frac{x}{q}}\right)}^p}\right)^{-\alpha -\beta}}{qB(\alpha,\beta)}</math>~~

	~~with [[mean]]~~

	~~: <math>\frac{q\Gamma(\alpha+\tfrac{1}{p})\Gamma(\beta-\tfrac{1}{p})}{\Gamma(\alpha)\Gamma(\beta)} \text{ if } \beta p>1</math>~~

	~~and~~ [~~[Mode (statistics)\|mode]]~~

	~~: <math>q{\left({\frac{\alpha p -1}{\beta p +1}}\right)}^\tfrac{1}{p} \text{ if } \alpha p\ge 1\!</math>~~

	~~Note that if p=q=1 then the generalized beta prime distribution reduces to the '''standard beta prime distribution'''~~

	~~===Compound gamma distribution===~~
	The '''compound gamma distribution'''<ref name=Dubey>{{cite journal\|last=Dubey\|first=Satya D.\|title=Compound gamma, beta and F distributions\|journal=Metrika\|date=December 1970\|volume=16\|pages=27–31\|doi=10.1007/BF02613934\|url=http://www.~~springerlink~~.com/~~content~~/~~u750hg4630387205~~/}}</ref> is the generalization of the beta prime when the scale parameter, ''q'' is added, but where ''p=1''. It is so named because it is formed by [[compound distribution\|compounding]] two [[gamma distribution]]s:

	:~~<math>\beta'(x;\alpha,\beta,1,q) = \int_0^\infty G(x;\alpha,p)G(p;\beta,q) \; dp<~~/~~math>~~

	~~where ''G(x;a,b)'' is the gamma distribution with shape ''a'' and ''inverse scale'' ''b''. This relationship can be used to generate random variables with a compound gamma, or beta prime distribution.~~

	~~The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by ''q'' and the variance by ''q<sup>2<~~/~~sup>''~~.

	~~==Properties==~~
	*If <math>X \sim \beta^{'}(\alpha,\beta)\,</math> then <math>\tfrac{1}{X} \sim \beta^{'}(\beta,\alpha)</math>.
	*If <math>X \sim \beta^{'}(\alpha,\beta,p,q)\,</math> then <math>kX \sim \beta^{'}(\alpha,\beta,p,kq)\,</math>.
	*<math>\beta^{'}(\alpha,\beta,1,1) = \beta^{'}(\alpha,\beta)\,</math>

	~~== Related distributions ==~~
	*If <math>X \sim F(\alpha,\beta)\,</math> then <math>\tfrac{\alpha}{\beta} X \sim \beta^{'}(\tfrac{\alpha}{2},\tfrac{\beta}{2})\,</math>
	*If <math>X \sim \textrm{Beta}(\alpha,\beta)\,</math> then <math>\frac{X}{1-X} \sim \beta^{'}(\alpha,\beta)\,</~~math>~~
	*If <math>X \sim \Gamma(\alpha,1)\,</~~math> and <math>Y \sim \Gamma(\beta,1)\,</math>, then <math>\frac{X}{Y} \sim \beta^{'}(\alpha,\beta)<~~/~~math>.~~
	*<math>\beta^{'}(p,1,a,b) = \textrm{Dagum}(p,a,b)\,</math> the [[Dagum distribution]]
	*<math>\beta^{'}(1,p,a,b) = \textrm{SinghMaddala}(p,a,b)\,</math> the [[Singh-Maddala distribution]]
	*<math>\beta^{'}(1,1,\gamma,\sigma) = \textrm{LL}(\gamma,\sigma)\,</math> the [[Log logistic distribution]]
	*Beta prime distribution is a special case of the type 6 [[Pearson distribution]]
	*[[Lomax distribution\|Pareto distribution type II]] is related to Beta prime distribution
	*[[Pareto distribution]] type IV is related to Beta prime distribution

	~~==Notes==~~
	~~{{Reflist}}~~

	~~==References==~~

	* Jonhnson, N.L., Kotz, S., Balakrishnan, N. (1995). ''Continuous Univariate Distributions'', Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
	* [http://mathworld.wolfram.com/BetaPrimeDistribution.html MathWorld article]

	~~{{ProbDistributions\|continuous-semi-infinite}}~~

	~~[[Category:Probability distributions]]~~
	~~[[Category:Continuous distributions]~~]

en>Monkbot: /* Bibliography */Fix CS1 deprecated date parameter errors

2014-01-22T03:13:52Z

Bibliography: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 05:13, 22 January 2014
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	~~std home test The writer~~ is known as ~~Irwin Wunder~~ [~~http:~~//~~www~~.~~blaze16~~.~~com~~/~~blog~~/~~257588 http~~://~~www~~.~~blaze16.com~~/~~blog~~/~~257588] but it~~'s ~~not~~ the ~~most masucline title out there~~. ~~His spouse doesn't like it~~ the ~~way he does but at home std test what he truly likes doing is~~ to ~~do aerobics and he~~'~~s been performing it for fairly a whilst~~. ~~Her spouse~~ and ~~her live in Puerto Rico but she [~~http://~~srgame~~.co.kr/~~qna~~/~~21862 srgame~~.co.~~kr] will have~~ to ~~move one day~~ or ~~another~~. ~~Managing individuals has been his day occupation for a whilst~~.<br><br>~~[http:~~//~~Rollingout~~.~~com~~/~~entertainment~~/~~celebrity~~-~~women~~-~~with-genital-herpes~~/ ~~Feel free~~] to ~~visit my website~~ ... [http://~~www~~.~~animecontent~~.com/~~blog/348813 home std test kit~~]		{{Probability distribution \|
			name =Beta Prime\|
			type =density\|
			pdf_image =[[Image:Beta prime pdf.svg\|325px]]\|
			cdf_image =[[Image:Beta prime cdf.svg\|325px]]\|
			parameters =<math>\alpha > 0</math> [[shape parameter\|shape]] ([[real number\|real]])<br /><math>\beta > 0</math> shape (real)\|
			support =<math>x > 0\!</math>\|
			pdf =<math>f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}\!</math>\|
			cdf =<math> I_{\frac{x}{1+x}(\alpha,\beta) }</math> where <math>I_x(\alpha,\beta)</math> is the incomplete beta function\|
			mean =<math>\frac{\alpha}{\beta-1} \text{ if } \beta>1</math>\|
			median =\|
			mode =<math>\frac{\alpha-1}{\beta+1} \text{ if } \alpha\ge 1\text{, 0 otherwise}\!</math>\|
			variance =<math>\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2} \text{ if } \beta>2</math>\|
			skewness =<math>\frac{2(2\alpha+\beta-1)}{\beta-3}\sqrt{\frac{\beta-2}{\alpha(\alpha+\beta-1)}} \text{ if } \beta>3</math>\|
			kurtosis =\|
			entropy =\|
			mgf =\|
			char =\|
			}}

			In [[probability theory]] and [[statistics]], the '''beta prime distribution''' (also known as '''inverted beta distribution''' or '''beta distribution of the second kind'''<ref name="Johnson et al 1995, p248">Johnson et al (1995), p248</ref>) is an [[probability distribution#Continuous probability distribution\|absolutely continuous probability distribution]] defined for <math>x > 0</math> with two parameters α and β, having the [[probability density function]]:

			: <math>f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}</math>

			where ''B'' is a [[Beta function]]. While the related [[beta distribution]] is the [[conjugate prior distribution]] of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in [[odds]]. The distribution is a [[Pearson distribution#The Pearson type VI distribution\|Pearson type VI]] distribution.<ref name="Johnson et al 1995, p248"/>

			The mode of a variate ''X'' distributed as <math>\beta^{'}(\alpha,\beta)</math> is <math>\hat{X} = \frac{\alpha-1}{\beta+1}</math>.
			Its mean is <math>\frac{\alpha}{\beta-1}</math> if <math>\beta>1</math> (if <math>\beta \leq 1</math> the mean is infinite, in other words it has no well defined mean)
			and its variance is
			<math>\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2}</math> if <math>\beta>2</math>.

			For <math>-\alpha <k <\beta </math>, the k-th moment <math> E[X^k] </math> is given by

			:<math> E[X^k]=\frac{B(\alpha+k,\beta-k)}{B(\alpha,\beta)}. </math>

			For <math> k\in \mathbb{N} </math> with <math>k <\beta </math>, this simplifies to

			:<math> E[X^k]=\prod_{i=1}^{k} \frac{\alpha+i-1}{\beta-i}. </math>

			The cdf can also be written as

			:<math> \frac{x^\alpha \cdot _2F_1(\alpha, \alpha+\beta, \alpha+1, -x)}{\alpha \cdot B(\alpha,\beta)}\!</math>
			where <math>_2F_1</math> is the Gauss's hypergeometric function <sub>2</sub>F<sub>1</sub> .

			==Generalization==

			Two more parameters can be added to form the '''generalized beta prime distribution'''.

			:<math>p > 0</math> [[shape parameter\|shape]] ([[real number\|real]]) <br> <math>q > 0</math> scale ([[real number\|real]])

			having the [[probability density function]]:

			: <math>f(x;\alpha,\beta,p,q) = \frac{p{\left({\frac{x}{q}}\right)}^{\alpha p-1} \left({1+{\left({\frac{x}{q}}\right)}^p}\right)^{-\alpha -\beta}}{qB(\alpha,\beta)}</math>

			with [[mean]]

			: <math>\frac{q\Gamma(\alpha+\tfrac{1}{p})\Gamma(\beta-\tfrac{1}{p})}{\Gamma(\alpha)\Gamma(\beta)} \text{ if } \beta p>1</math>

			and [[Mode (statistics)\|mode]]

			: <math>q{\left({\frac{\alpha p -1}{\beta p +1}}\right)}^\tfrac{1}{p} \text{ if } \alpha p\ge 1\!</math>

			Note that if p=q=1 then the generalized beta prime distribution reduces to the '''standard beta prime distribution'''

			===Compound gamma distribution===
			The '''compound gamma distribution'''<ref name=Dubey>{{cite journal\|last=Dubey\|first=Satya D.\|title=Compound gamma, beta and F distributions\|journal=Metrika\|date=December 1970\|volume=16\|pages=27–31\|doi=10.1007/BF02613934\|url=http://www.springerlink.com/content/u750hg4630387205/}}</ref> is the generalization of the beta prime when the scale parameter, ''q'' is added, but where ''p=1''. It is so named because it is formed by [[compound distribution\|compounding]] two [[gamma distribution]]s:

			:<math>\beta'(x;\alpha,\beta,1,q) = \int_0^\infty G(x;\alpha,p)G(p;\beta,q) \; dp</math>

			where ''G(x;a,b)'' is the gamma distribution with shape ''a'' and ''inverse scale'' ''b''. This relationship can be used to generate random variables with a compound gamma, or beta prime distribution.

			The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by ''q'' and the variance by ''q<sup>2</sup>''.

			==Properties==
			*If <math>X \sim \beta^{'}(\alpha,\beta)\,</math> then <math>\tfrac{1}{X} \sim \beta^{'}(\beta,\alpha)</math>.
			*If <math>X \sim \beta^{'}(\alpha,\beta,p,q)\,</math> then <math>kX \sim \beta^{'}(\alpha,\beta,p,kq)\,</math>.
			*<math>\beta^{'}(\alpha,\beta,1,1) = \beta^{'}(\alpha,\beta)\,</math>

			== Related distributions ==
			*If <math>X \sim F(\alpha,\beta)\,</math> then <math>\tfrac{\alpha}{\beta} X \sim \beta^{'}(\tfrac{\alpha}{2},\tfrac{\beta}{2})\,</math>
			*If <math>X \sim \textrm{Beta}(\alpha,\beta)\,</math> then <math>\frac{X}{1-X} \sim \beta^{'}(\alpha,\beta)\,</math>
			*If <math>X \sim \Gamma(\alpha,1)\,</math> and <math>Y \sim \Gamma(\beta,1)\,</math>, then <math>\frac{X}{Y} \sim \beta^{'}(\alpha,\beta)</math>.
			*<math>\beta^{'}(p,1,a,b) = \textrm{Dagum}(p,a,b)\,</math> the [[Dagum distribution]]
			*<math>\beta^{'}(1,p,a,b) = \textrm{SinghMaddala}(p,a,b)\,</math> the [[Singh-Maddala distribution]]
			*<math>\beta^{'}(1,1,\gamma,\sigma) = \textrm{LL}(\gamma,\sigma)\,</math> the [[Log logistic distribution]]
			*Beta prime distribution is a special case of the type 6 [[Pearson distribution]]
			*[[Lomax distribution\|Pareto distribution type II]] is related to Beta prime distribution
			*[[Pareto distribution]] type IV is related to Beta prime distribution

			==Notes==
			{{Reflist}}

			==References==

			* Jonhnson, N.L., Kotz, S., Balakrishnan, N. (1995). ''Continuous Univariate Distributions'', Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
			* [http://mathworld.wolfram.com/BetaPrimeDistribution.html MathWorld article]

			{{ProbDistributions\|continuous-semi-infinite}}

			[[Category:Probability distributions]]
			[[Category:Continuous distributions]]

en>TheCois: Added an author link

2012-08-24T18:42:22Z

Added an author link

New page

std home test The writer is known as Irwin Wunder [http://www.blaze16.com/blog/257588 http://www.blaze16.com/blog/257588] but it's not the most masucline title out there. His spouse doesn't like it the way he does but at home std test what he truly likes doing is to do aerobics and he's been performing it for fairly a whilst. Her spouse and her live in Puerto Rico but she [http://srgame.co.kr/qna/21862 srgame.co.kr] will have to move one day or another. Managing individuals has been his day occupation for a whilst.<br><br>[http://Rollingout.com/entertainment/celebrity-women-with-genital-herpes/ Feel free] to visit my website ... [http://www.animecontent.com/blog/348813 home std test kit]