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| {{Probability distribution|
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| name =Scaled inverse chi-squared|
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| type =density|
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| pdf_image =[[File:Scaled inverse chi squared.svg|250px]]|
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| cdf_image =[[File:Scaled inverse chi squared cdf.svg|250px]]|
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| parameters =<math>\nu > 0\,</math><BR /><math>\tau^2 > 0\,</math> |
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| support =<math>x \in (0, \infty)</math>|
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| pdf =<math>\frac{(\tau^2\nu/2)^{\nu/2}}{\Gamma(\nu/2)}~
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| \frac{\exp\left[ \frac{-\nu \tau^2}{2 x}\right]}{x^{1+\nu/2}} </math>|
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| cdf =<math>\Gamma\left(\frac{\nu}{2},\frac{\tau^2\nu}{2x}\right)
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| \left/\Gamma\left(\frac{\nu}{2}\right)\right.</math>|
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| mean =<math>\frac{\nu \tau^2}{\nu-2}</math> for <math>\nu >2\,</math>|
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| median =|
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| mode =<math>\frac{\nu \tau^2}{\nu+2}</math>|
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| variance =<math>\frac{2 \nu^2 \tau^4}{(\nu-2)^2 (\nu-4)}</math>for <math>\nu >4\,</math>|
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| skewness =<math>\frac{4}{\nu-6}\sqrt{2(\nu-4)}</math>for <math>\nu >6\,</math>|
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| kurtosis =<math>\frac{12(5\nu-22)}{(\nu-6)(\nu-8)}</math>for <math>\nu >8\,</math>|
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| entropy =<math>\frac{\nu}{2}
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| \!+\!\ln\left(\frac{\tau^2\nu}{2}\Gamma\left(\frac{\nu}{2}\right)\right)</math>
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| <math>\!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\left(\frac{\nu}{2}\right)</math>|
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| mgf =<math>\frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-\tau^2\nu t}{2}\right)^{\!\!\frac{\nu}{4}}\!\!K_{\frac{\nu}{2}}\left(\sqrt{-2\tau^2\nu t}\right)</math>|
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| char =<math>\frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-i\tau^2\nu t}{2}\right)^{\!\!\frac{\nu}{4}}\!\!K_{\frac{\nu}{2}}\left(\sqrt{-2i\tau^2\nu t}\right)</math>|
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| }}
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| The '''scaled inverse chi-squared distribution''' is the distribution for ''x'' = 1/''s''<sup>2</sup>, where ''s''<sup>2</sup> is a sample mean of the squares of ν independent [[normal distribution|normal]] random variables that have mean 0 and inverse variance 1/σ<sup>2</sup> = τ<sup>2</sup>. The distribution is therefore parametrised by the two quantities ν and τ<sup>2</sup>, referred to as the ''number of chi-squared degrees of freedom'' and the ''scaling parameter'', respectively. | |
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| This family of scaled inverse chi-squared distributions is closely related to two other distribution families, those of the [[inverse-chi-squared distribution]] and the [[inverse gamma distribution]]. Compared to the inverse-chi-squared distribution, the scaled distribution has an extra parameter ''τ''<sup>2</sup>, which scales the distribution horizontally and vertically, representing the inverse-variance of the original underlying process. Also, the scale inverse chi-squared distribution is presented as the distribution for the inverse of the ''mean'' of ν squared deviates, rather than the inverse of their ''sum''. The two distributions thus have the relation that if
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| :<math>X \sim \mbox{Scale-inv-}\chi^2(\nu, \tau^2)</math> then <math> \frac{X}{\tau^2 \nu} \sim \mbox{inv-}\chi^2(\nu)</math> | |
| Compared to the inverse gamma distribution, the scaled inverse chi-squared distribution describes the same data distribution, but using a different [[parametrization]], which may be more convenient in some circumstances. Specifically, if
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| :<math>X \sim \mbox{Scale-inv-}\chi^2(\nu, \tau^2)</math> then <math>X \sim \textrm{Inv-Gamma}\left(\frac{\nu}{2}, \frac{\nu\tau^2}{2}\right)</math>
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| Either form may be used to represent the [[maximum entropy]] distribution for a fixed first inverse [[Moment (mathematics)|moment]] <math>(E(1/X))</math> and first logarithmic moment <math>(E(\ln(X))</math>.
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| The scaled inverse chi-squared distribution also has a particular use in [[Bayesian statistics]], somewhat unrelated to its use as a predictive distribution for ''x'' = 1/''s''<sup>2</sup>. Specifically, the scaled inverse chi-squared distribution can be used as a [[conjugate prior]] for the [[variance]] parameter of a [[normal distribution]]. In this context the scaling parameter is denoted by σ<sub>0</sub><sup>2</sup> rather than by τ<sup>2</sup>, and has a different interpretation. The application has been more usually presented using the [[inverse gamma distribution]] formulation instead; however, some authors, following in particular Gelman ''et al.'' (1995/2004) argue that the inverse chi-squared parametrisation is more intuitive.
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| ==Characterization==
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| The [[probability density function]] of the scaled inverse chi-squared distribution extends over the domain <math>x>0</math> and is
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| :<math>
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| f(x; \nu, \tau^2)=
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| \frac{(\tau^2\nu/2)^{\nu/2}}{\Gamma(\nu/2)}~
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| \frac{\exp\left[ \frac{-\nu \tau^2}{2 x}\right]}{x^{1+\nu/2}}
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| </math>
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| where <math>\nu</math> is the [[degrees of freedom (statistics)|degrees of freedom]] parameter and <math>\tau^2</math> is the [[scale parameter]]. The cumulative distribution function is
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| :<math>F(x; \nu, \tau^2)=
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| \Gamma\left(\frac{\nu}{2},\frac{\tau^2\nu}{2x}\right)
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| \left/\Gamma\left(\frac{\nu}{2}\right)\right.</math>
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| :<math>=Q\left(\frac{\nu}{2},\frac{\tau^2\nu}{2x}\right)</math>
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| where <math>\Gamma(a,x)</math> is the [[incomplete Gamma function]], <math>\Gamma(x)</math> is the [[Gamma function]] and <math>Q(a,x)</math> is a [[incomplete Gamma function|regularized Gamma function]]. The [[Characteristic function (probability theory)|characteristic function]] is
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| :<math>\varphi(t;\nu,\tau^2)=</math>
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| :<math>\frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-i\tau^2\nu t}{2}\right)^{\!\!\frac{\nu}{4}}\!\!K_{\frac{\nu}{2}}\left(\sqrt{-2i\tau^2\nu t}\right) ,</math>
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| where <math>K_{\frac{\nu}{2}}(z)</math> is the modified [[Bessel function]] of the second kind.
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| ==Parameter estimation==
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| The [[maximum likelihood]] estimate of <math>\tau^2</math> is
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| :<math>\tau^2 = n/\sum_{i=1}^n \frac{1}{x_i}.</math>
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| The [[maximum likelihood]] estimate of <math>\frac{\nu}{2}</math> can be found using [[Newton's method]] on:
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| :<math>\ln(\frac{\nu}{2}) + \psi(\frac{\nu}{2}) = \sum_{i=1}^n \ln(x_i) - n \ln(\tau^2) ,</math>
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| where <math>\psi(x)</math> is the [[digamma function]]. An initial estimate can be found by taking the formula for mean and solving it for <math>\nu.</math> Let <math>\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i</math> be the sample mean. Then an initial estimate for <math>\nu</math> is given by:
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| :<math>\frac{\nu}{2} = \frac{\bar{x}}{\bar{x} - \tau^2}.</math>
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| ==Bayesian estimation of the variance of a Normal distribution==
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| The scaled inverse chi-squared distribution has a second important application, in the Bayesian estimation of the variance of a Normal distribution.
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| According to [[Bayes theorem]], the [[posterior probability distribution]] for quantities of interest is proportional to the product of a [[prior distribution]] for the quantities and a [[likelihood function]]:
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| :<math>p(\sigma^2|D,I) \propto p(\sigma^2|I) \; p(D|\sigma^2)</math>
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| where ''D'' represents the data and ''I'' represents any initial information about σ<sup>2</sup> that we may already have.
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| The simplest scenario arises if the mean μ is already known; or, alternatively, if it is the [[conditional probability|conditional distribution]] of σ<sup>2</sup> that is sought, for a particular assumed value of μ.
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| Then the likelihood term ''L''(σ<sup>2</sup>|''D'') = ''p''(''D''|σ<sup>2</sup>) has the familiar form
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| :<math>\mathcal{L}(\sigma^2|D,\mu) = \frac{1}{\left(\sqrt{2\pi}\sigma\right)^n} \; \exp \left[ -\frac{\sum_i^n(x_i-\mu)^2}{2\sigma^2} \right]</math>
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| Combining this with the rescaling-invariant prior p(σ<sup>2</sup>|''I'') = 1/σ<sup>2</sup>, which can be argued (e.g. [[Jeffreys_prior#Gaussian_distribution_with_standard_deviation_parameter|following Jeffreys]]) to be the least informative possible prior for σ<sup>2</sup> in this problem, gives a combined posterior probability
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| :<math>p(\sigma^2|D, I, \mu) \propto \frac{1}{\sigma^{n+2}} \; \exp \left[ -\frac{\sum_i^n(x_i-\mu)^2}{2\sigma^2} \right]</math> | |
| This form can be recognised as that of a scaled inverse chi-squared distribution, with parameters ν = ''n'' and τ<sup>2</sup> = ''s''<sup>2</sup> = (1/''n'') Σ (x<sub>i</sub>-μ)<sup>2</sup>
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| Gelman ''et al'' remark that the re-appearance of this distribution, previously seen in a sampling context, may seem remarkable; but given the choice of prior the "result is not surprising".<ref>Gelman ''et al'' (1995), ''Bayesian Data Analysis'' (1st ed), p.68</ref>
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| In particular, the choice of a rescaling-invariant prior for σ<sup>2</sup> has the result that the probability for the ratio of σ<sup>2</sup> / ''s''<sup>2</sup> has the same form (independent of the conditioning variable) when conditioned on ''s''<sup>2</sup> as when conditioned on σ<sup>2</sup>:
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| :<math>p(\tfrac{\sigma^2}{s^2}|s^2) = p(\tfrac{\sigma^2}{s^2}|\sigma^2)</math>
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| In the sampling-theory case, conditioned on σ<sup>2</sup>, the probability distribution for (1/s<sup>2</sup>) is a scaled inverse chi-squared distribution; and so the probability distribution for σ<sup>2</sup> conditioned on ''s''<sup>2</sup>, given a scale-agnostic prior, is also a scaled inverse chi-squared distribution.
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| === Use as an informative prior ===
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| If more is known about the possible values of σ<sup>2</sup>, a distribution from the scaled inverse chi-squared family, such as Scale-inv-χ<sup>2</sup>(''n''<sub>0</sub>, ''s''<sub>0</sub><sup>2</sup>) can be a convenient form to represent a less uninformative prior for σ<sup>2</sup>, as if from the result of ''n''<sub>0</sub> previous observations (though ''n''<sub>0</sub> need not necessarily be a whole number):
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| :<math>p(\sigma^2|I^\prime, \mu) \propto \frac{1}{\sigma^{n_0+2}} \; \exp \left[ -\frac{n_0 s_0^2}{2\sigma^2} \right]</math>
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| Such a prior would lead to the posterior distribution
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| :<math>p(\sigma^2|D, I^\prime, \mu) \propto \frac{1}{\sigma^{n+n_0+2}} \; \exp \left[ -\frac{\sum{ns^2 + n_0 s_0^2}}{2\sigma^2} \right]</math>
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| which is itself a scaled inverse chi-squared distribution. The scaled inverse chi-squared distributions are thus a convenient [[conjugate prior]] family for σ<sup>2</sup> estimation.
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| === Estimation of variance when mean is unknown ===
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| If the mean is not known, the most uninformative prior that can be taken for it is arguably the translation-invariant prior ''p''(μ|''I'') ∝ const., which gives the following joint posterior distribution for μ and σ<sup>2</sup>,
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| :<math>\begin{align}
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| p(\mu, \sigma^2|D, I) \; \propto \; & \frac{1}{\sigma^{n+2}} \; \exp \left[ -\frac{\sum_i^n(x_i-\mu)^2}{2\sigma^2} \right] \\
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| = \; & \frac{1}{\sigma^{n+2}} \; \exp \left[ -\frac{\sum_i^n(x_i-\bar{x})^2}{2\sigma^2} \right] \; \exp \left[ -\frac{\sum_i^n(\mu -\bar{x})^2}{2\sigma^2} \right]
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| \end{align}</math>
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| The marginal posterior distribution for σ<sup>2</sup> is obtained from the joint posterior distribution by integrating out over μ,
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| :<math>\begin{align}
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| p(\sigma^2|D, I) \; \propto \; & \frac{1}{\sigma^{n+2}} \; \exp \left[ -\frac{\sum_i^n(x_i-\bar{x})^2}{2\sigma^2} \right] \; \int_{-\infty}^{\infty} \exp \left[ -\frac{\sum_i^n(\mu -\bar{x})^2}{2\sigma^2} \right] d\mu\\
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| = \; & \frac{1}{\sigma^{n+2}} \; \exp \left[ -\frac{\sum_i^n(x_i-\bar{x})^2}{2\sigma^2} \right] \; \sqrt{2 \pi \sigma^2 / n} \\
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| \propto \; & (\sigma^2)^{-(n+1)/2} \; \exp \left[ -\frac{(n-1)s^2}{2\sigma^2} \right]
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| \end{align}</math>
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| This is again a scaled inverse chi-squared distribution, with parameters <math>\scriptstyle{n-1}\;</math> and <math>\scriptstyle{s^2 = \sum (x_i - \bar{x})^2/(n-1)}</math>.
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| ==Related distributions==
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| * If <math>X \sim \mbox{Scale-inv-}\chi^2(\nu, \tau^2)</math> then <math> k X \sim \mbox{Scale-inv-}\chi^2(\nu, k \tau^2)\, </math>
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| * If <math>X \sim \mbox{inv-}\chi^2(\nu) \, </math> ([[Inverse-chi-squared distribution]]) then <math>X \sim \mbox{Scale-inv-}\chi^2(\nu, 1/\nu) \,</math>
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| * If <math>X \sim \mbox{Scale-inv-}\chi^2(\nu, \tau^2)</math> then <math> \frac{X}{\tau^2 \nu} \sim \mbox{inv-}\chi^2(\nu) \, </math> ([[Inverse-chi-squared distribution]])
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| * If <math>X \sim \mbox{Scale-inv-}\chi^2(\nu, \tau^2)</math> then <math>X \sim \textrm{Inv-Gamma}\left(\frac{\nu}{2}, \frac{\nu\tau^2}{2}\right)</math> ([[Inverse-gamma distribution]])
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| * Scaled inverse chi square distribution is a special case of type 5 [[Pearson distribution]]
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| == References ==
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| * Gelman A. ''et al'' (1995), ''Bayesian Data Analysis'', pp 474–475; also pp 47, 480
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| {{reflist}}
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{DEFAULTSORT:Scaled-Inverse-Chi-Squared Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Exponential family distributions]]
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| [[Category:Probability distributions]]
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