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Beta distribution
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====Median==== [[File:Median Beta Distribution for alpha and beta from 0 to 5 - J. Rodal.jpg|325px|thumb|Median for beta distribution for 0 ≤ ''α'' ≤ 5 and 0 ≤ ''β'' ≤ 5]] [[File:(Mean - Median) for Beta distribution versus alpha and beta from 0 to 2 - J. Rodal.jpg|thumb|(Mean–median) for beta distribution versus alpha and beta from 0 to 2]] The median of the beta distribution is the unique real number <math>x = I_{1/2}^{[-1]}(\alpha,\beta)</math> for which the [[regularized incomplete beta function]] <math>I_x(\alpha,\beta) = \tfrac{1}{2} </math>. There is no general [[closed-form expression]] for the [[median]] of the beta distribution for arbitrary values of ''α'' and ''β''. [[Closed-form expression]]s for particular values of the parameters ''α'' and ''β'' follow:{{citation needed|date=February 2013}} * For symmetric cases ''α'' = ''β'', median = 1/2. * For ''α'' = 1 and ''β'' > 0, median <math> =1-2^{-1/\beta}</math> (this case is the [[mirror image|mirror-image]] of the power function [0,1] distribution) * For ''α'' > 0 and ''β'' = 1, median = <math>2^{-1/\alpha}</math> (this case is the power function [0,1] distribution<ref name="Handbook of Beta Distribution" />) * For ''α'' = 3 and ''β'' = 2, median = 0.6142724318676105..., the real solution to the [[Quartic function|quartic equation]] 1 − 8''x''<sup>3</sup> + 6''x''<sup>4</sup> = 0, which lies in [0,1]. * For ''α'' = 2 and ''β'' = 3, median = 0.38572756813238945... = 1−median(Beta(3, 2)) The following are the limits with one parameter finite (non-zero) and the other approaching these limits:{{citation needed|date=February 2013}} :<math> \begin{align} \lim_{\beta \to 0} \text{median}= \lim_{\alpha \to \infty} \text{median} = 1,\\ \lim_{\alpha\to 0} \text{median}= \lim_{\beta \to \infty} \text{median} = 0. \end{align}</math> A reasonable approximation of the value of the median of the beta distribution, for both α and β greater or equal to one, is given by the formula<ref name=Kerman2011/> :<math>\text{median} \approx \frac{\alpha - \tfrac{1}{3}}{\alpha + \beta - \tfrac{2}{3}} \text{ for } \alpha, \beta \ge 1.</math> When ''α'', ''β'' ≥ 1, the [[relative error]] (the [[approximation error|absolute error]] divided by the median) in this approximation is less than 4% and for both ''α'' ≥ 2 and ''β'' ≥ 2 it is less than 1%. The [[approximation error|absolute error]] divided by the difference between the mean and the mode is similarly small: [[File:Relative Error for Approximation to Median of Beta Distribution for alpha and beta from 1 to 5 - J. Rodal.jpg|325px|Abs[(Median-Appr.)/Median] for beta distribution for 1 ≤ ''α'' ≤ 5 and 1 ≤ ''β'' ≤ 5]][[File:Error in Median Apprx. relative to Mean-Mode distance for Beta Distribution with alpha and beta from 1 to 5 - J. Rodal.jpg|325px|Abs[(Median-Appr.)/(Mean-Mode)] for beta distribution for 1 ≤ ''α'' ≤ 5 and 1 ≤ ''β'' ≤ 5]]
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