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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|
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| type =density|
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| pdf_image =[[Image:Beta prime pdf.svg|325px]]|
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| cdf_image =[[Image:Beta prime cdf.svg|325px]]|
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| parameters =<math>\alpha > 0</math> [[shape parameter|shape]] ([[real number|real]])<br /><math>\beta > 0</math> shape (real)|
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| support =<math>x > 0\!</math>|
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| pdf =<math>f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}\!</math>|
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| cdf =<math> I_{\frac{x}{1+x}(\alpha,\beta) }</math> where <math>I_x(\alpha,\beta)</math> is the incomplete beta function|
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| mean =<math>\frac{\alpha}{\beta-1} \text{ if } \beta>1</math>|
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| median =|
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| mode =<math>\frac{\alpha-1}{\beta+1} \text{ if } \alpha\ge 1\text{, 0 otherwise}\!</math>|
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| variance =<math>\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2} \text{ if } \beta>2</math>|
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| skewness =<math>\frac{2(2\alpha+\beta-1)}{\beta-3}\sqrt{\frac{\beta-2}{\alpha(\alpha+\beta-1)}} \text{ if } \beta>3</math>|
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| kurtosis =|
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| entropy =|
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| mgf =|
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| char =|
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| }}
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| 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]]:
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| : <math>f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}</math>
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| 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"/>
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| The mode of a variate ''X'' distributed as <math>\beta^{'}(\alpha,\beta)</math> is <math>\hat{X} = \frac{\alpha-1}{\beta+1}</math>.
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| 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)
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| and its variance is | |
| <math>\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2}</math> if <math>\beta>2</math>.
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| For <math>-\alpha <k <\beta </math>, the k-th moment <math> E[X^k] </math> is given by
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| :<math> E[X^k]=\frac{B(\alpha+k,\beta-k)}{B(\alpha,\beta)}. </math>
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| For <math> k\in \mathbb{N} </math> with <math>k <\beta </math>, this simplifies to
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| :<math> E[X^k]=\prod_{i=1}^{k} \frac{\alpha+i-1}{\beta-i}. </math>
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| The cdf can also be written as
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| :<math> \frac{x^\alpha \cdot _2F_1(\alpha, \alpha+\beta, \alpha+1, -x)}{\alpha \cdot B(\alpha,\beta)}\!</math>
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| where <math>_2F_1</math> is the Gauss's hypergeometric function <sub>2</sub>F<sub>1</sub> . | |
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| ==Generalization==
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| Two more parameters can be added to form the '''generalized beta prime distribution'''.
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| :<math>p > 0</math> [[shape parameter|shape]] ([[real number|real]]) <br> <math>q > 0</math> scale ([[real number|real]])
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| having the [[probability density function]]:
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| : <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> | |
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| with [[mean]]
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| : <math>\frac{q\Gamma(\alpha+\tfrac{1}{p})\Gamma(\beta-\tfrac{1}{p})}{\Gamma(\alpha)\Gamma(\beta)} \text{ if } \beta p>1</math>
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| and [[Mode (statistics)|mode]]
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| : <math>q{\left({\frac{\alpha p -1}{\beta p +1}}\right)}^\tfrac{1}{p} \text{ if } \alpha p\ge 1\!</math>
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| Note that if p=q=1 then the generalized beta prime distribution reduces to the '''standard beta prime distribution'''
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| ===Compound gamma distribution===
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| 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:
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| :<math>\beta'(x;\alpha,\beta,1,q) = \int_0^\infty G(x;\alpha,p)G(p;\beta,q) \; dp</math> | |
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| 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.
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| 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>''.
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| ==Properties==
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| *If <math>X \sim \beta^{'}(\alpha,\beta)\,</math> then <math>\tfrac{1}{X} \sim \beta^{'}(\beta,\alpha)</math>.
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| *If <math>X \sim \beta^{'}(\alpha,\beta,p,q)\,</math> then <math>kX \sim \beta^{'}(\alpha,\beta,p,kq)\,</math>.
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| *<math>\beta^{'}(\alpha,\beta,1,1) = \beta^{'}(\alpha,\beta)\,</math>
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| == Related distributions ==
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| *If <math>X \sim F(\alpha,\beta)\,</math> then <math>\tfrac{\alpha}{\beta} X \sim \beta^{'}(\tfrac{\alpha}{2},\tfrac{\beta}{2})\,</math>
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| *If <math>X \sim \textrm{Beta}(\alpha,\beta)\,</math> then <math>\frac{X}{1-X} \sim \beta^{'}(\alpha,\beta)\,</math>
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| *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>.
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| *<math>\beta^{'}(p,1,a,b) = \textrm{Dagum}(p,a,b)\,</math> the [[Dagum distribution]]
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| *<math>\beta^{'}(1,p,a,b) = \textrm{SinghMaddala}(p,a,b)\,</math> the [[Singh-Maddala distribution]]
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| *<math>\beta^{'}(1,1,\gamma,\sigma) = \textrm{LL}(\gamma,\sigma)\,</math> the [[Log logistic distribution]]
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| *Beta prime distribution is a special case of the type 6 [[Pearson distribution]]
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| *[[Lomax distribution|Pareto distribution type II]] is related to Beta prime distribution
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| *[[Pareto distribution]] type IV is related to Beta prime distribution
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * Jonhnson, N.L., Kotz, S., Balakrishnan, N. (1995). ''Continuous Univariate Distributions'', Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
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| * [http://mathworld.wolfram.com/BetaPrimeDistribution.html MathWorld article]
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| {{ProbDistributions|continuous-semi-infinite}}
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| [[Category:Probability distributions]]
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| [[Category:Continuous distributions]]
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