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| {{Probability distribution
| | Hello, I'm Clifton, a 21 year old from Cavaillon, France.<br>My hobbies include (but are not limited to) Gaming, Magic and watching Grey's Anatomy.<br><br>My web-site - [http://tinyurl.com/q6d8aya Cheap ghd uk] |
| | name = Beta Negative Binomial
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| | type = mass
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| | pdf_image = No image available
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| | cdf_image = No image available
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| | notation =
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| | parameters = <math>\alpha > 0</math> [[shape parameter|shape]] ([[real number|real]])<br /><math>\beta > 0</math> [[shape parameter|shape]] ([[real number|real]]) <br> ''n'' ∈ [[Natural numbers|'''N'''<sub>0</sub>]] — number of trials
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| | support = ''k'' ∈ { 0, 1, 2, 3, ... }
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| | pdf = <math>\frac{n^{(k)}\alpha^{(n)}\beta^{(k)}}{k!(\alpha+\beta)^{(n)}(n+\alpha+\beta)^{(k)}}</math><br>Where <math>x^{(n)}</math> is the '''rising''' [[Pochhammer symbol]]
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| | cdf =
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| | mean = <math>\begin{cases}
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| \frac{n\beta}{\alpha-1} & \text{if}\ \alpha>1 \\
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| \infty & \text{otherwise}\ \end{cases}</math>
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| | median =
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| | mode =
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| | variance = <math>\begin{cases}
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| \frac{n(\alpha+n-1)\beta(\alpha+\beta-1)}{(\alpha-2){(\alpha-1)}^2} & \text{if}\ \alpha>2 \\
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| \infty & \text{otherwise}\ \end{cases}</math>
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| | skewness = <math>\begin{cases}
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| \frac{(\alpha+2n-1)(\alpha+2\beta-1)}{(\alpha-3)\sqrt{\frac{n(\alpha+n-1)\beta(\alpha+\beta-1)}{\alpha-2}}} & \text{if}\ \alpha>3 \\
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| \infty & \text{otherwise}\ \end{cases}</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]], a '''beta negative binomial distribution''' is the [[probability distribution]] of a [[discrete probability distribution|discrete]] [[random variable]] ''X'' equal to the number of failures needed to get ''n'' successes in a sequence of [[independence (probability theory)|independent]] [[Bernoulli trial]]s where the probability ''p'' of success on each trial is constant within any given experiment but is itself a random variable following a [[beta distribution]], varying between different experiments. Thus the distribution is a [[compound probability distribution]].
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| This distribution has also been called both the '''inverse Markov-Pólya distribution''' and the '''generalized Waring distribution'''.<ref name=Johnson>Johnson et al. (1993)</ref> A shifted form of the distribution has been called the '''beta-Pascal distribution'''.<ref name=Johnson/>
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| If parameters of the beta distribution are ''α'' and ''β'' , and if
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| :<math>
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| X \mid p \sim \mathrm{NB}(n,p),
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| </math>
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| where
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| :<math>
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| p \sim \textrm{B}(\alpha,\beta),
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| </math>
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| then the marginal distribution of ''X'' is a beta negative binomial distribution:
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| :<math>
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| X \sim \mathrm{BNB}(n,\alpha,\beta).
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| </math>
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| In the above, NB(''n'', ''p'') is the [[negative binomial distribution]] and B(''α'', ''β'') is the [[beta distribution]].
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| ==PMF expressed with Gamma==
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| Since the rising Pochhammer symbol can be expressed in terms of the [[Gamma function]], the numerator of the PMF as given can be expressed as:
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| :<math>\frac{\Gamma(n+k)\Gamma(\alpha+n)\Gamma(\beta+k)}{\Gamma(n)\Gamma(\alpha)\Gamma(\beta)}</math>.
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| Likewise, the denominator can be rewritten as:
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| :<math>\frac{\Gamma(\alpha+\beta)\Gamma(\alpha+\beta+n)}{k!\Gamma(\alpha+\beta+n)\Gamma(\alpha+\beta+n+k)}</math>,
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| and the two <math>{\Gamma(\alpha+\beta+n)}</math> terms cancel out, leaving:
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| :<math>\frac{\Gamma(\alpha+\beta)}{k!\Gamma(\alpha+\beta+n+k)}</math>.
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| As <math>\frac{\Gamma(n+k)}{k!\Gamma(n)} = \binom{n+k-1}k</math>, the PMF can be rewritten as:
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| :<math>\binom{n+k-1}k\frac{\Gamma(\alpha+n)\Gamma(\beta+k)\Gamma(\alpha+\beta)}{\Gamma(\alpha+\beta+n+k)\Gamma(\alpha)\Gamma(\beta)}</math>.
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| ===PMF expressed with Beta===
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| Using the [[beta function]], the PMF is:
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| :<math>\binom{n+k-1}k\frac{\Beta(\alpha+n,\beta+k)}{\Beta(\alpha,\beta)}</math>.
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| Replacing the [[binomial coefficient]] by a beta function, the PMF can also be written:
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| :<math>\frac{\Beta(\alpha+n,\beta+k)}{k\Beta(\alpha,\beta)\Beta(n,k)}</math>.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *Jonhnson, N.L.; Kotz, S.; Kemp, A.W. (1993) ''Univariate Discrete Distributions'', 2nd edition, Wiley ISBN 0-471-54897-9 (Section 6.2.3)
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| *Kemp, C.D.; Kemp, A.W. (1956) "Generalized hypergeometric distributions'', ''[[Journal of the Royal Statistical Society]]'', Series B, 18, 202–211
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| *Wang, Zhaoliang (2011) "One mixed negative binomial distribution with application", ''Journal of Statistical Planning and Inference'', 141 (3), 1153-1160 {{DOI|10.1016/j.jspi.2010.09.020}}
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| ==External links==
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| * Interactive graphic: [http://www.math.wm.edu/~leemis/chart/UDR/UDR.html Univariate Distribution Relationships]
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| {{ProbDistributions|discrete-infinite}}
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| [[Category:Discrete distributions]]
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| [[Category:Compound distributions]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Probability distributions]]
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Hello, I'm Clifton, a 21 year old from Cavaillon, France.
My hobbies include (but are not limited to) Gaming, Magic and watching Grey's Anatomy.
My web-site - Cheap ghd uk