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Beta distribution
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===Geometry of the probability density function=== ====Inflection points==== [[File:Inflexion points Beta Distribution alpha and beta ranging from 0 to 5 large ptl view - J. Rodal.jpg|thumb|Inflection point location versus α and β showing regions with one inflection point]] [[File:Inflexion points Beta Distribution alpha and beta ranging from 0 to 5 large ptr view - J. Rodal.jpg|thumb|Inflection point location versus α and β showing region with two inflection points]] For certain values of the shape parameters α and β, the [[probability density function]] has [[inflection points]], at which the [[curvature]] changes sign. The position of these inflection points can be useful as a measure of the [[Statistical dispersion|dispersion]] or spread of the distribution. Defining the following quantity: :<math>\kappa =\frac{\sqrt{\frac{(\alpha-1)(\beta-1)}{\alpha+\beta-3}}}{\alpha+\beta-2}</math> Points of inflection occur,<ref name=JKB /><ref name=Wadsworth /><ref name="Handbook of Beta Distribution" /><ref name=Panik /> depending on the value of the shape parameters ''α'' and ''β'', as follows: *(''α'' > 2, ''β'' > 2) The distribution is bell-shaped (symmetric for ''α'' = ''β'' and skewed otherwise), with '''two inflection points''', equidistant from the mode: ::<math>x = \text{mode} \pm \kappa = \frac{\alpha -1 \pm \sqrt{\frac{(\alpha-1)(\beta-1)}{\alpha+\beta-3}}}{\alpha+\beta-2}</math> * (''α'' = 2, ''β'' > 2) The distribution is unimodal, positively skewed, right-tailed, with '''one inflection point''', located to the right of the mode: ::<math>x =\text{mode} + \kappa = \frac{2}{\beta}</math> * (''α'' > 2, β = 2) The distribution is unimodal, negatively skewed, left-tailed, with '''one inflection point''', located to the left of the mode: ::<math>x = \text{mode} - \kappa = 1 - \frac{2}{\alpha}</math> * (1 < ''α'' < 2, β > 2, ''α'' + ''β'' > 2) The distribution is unimodal, positively skewed, right-tailed, with '''one inflection point''', located to the right of the mode: ::<math>x =\text{mode} + \kappa = \frac{\alpha -1 +\sqrt{\frac{(\alpha-1)(\beta-1)}{\alpha+\beta-3}}}{\alpha+\beta-2}</math> *(0 < ''α'' < 1, 1 < ''β'' < 2) The distribution has a mode at the left end ''x'' = 0 and it is positively skewed, right-tailed. There is '''one inflection point''', located to the right of the mode: ::<math>x = \frac{\alpha -1 +\sqrt{\frac{(\alpha-1)(\beta-1)}{\alpha+\beta-3}}}{\alpha+\beta-2}</math> *(''α'' > 2, 1 < ''β'' < 2) The distribution is unimodal negatively skewed, left-tailed, with '''one inflection point''', located to the left of the mode: ::<math>x =\text{mode} - \kappa = \frac{\alpha -1 -\sqrt{\frac{(\alpha-1)(\beta-1)}{\alpha+\beta-3}}}{\alpha+\beta-2}</math> *(1 < ''α'' < 2, 0 < ''β'' < 1) The distribution has a mode at the right end ''x'' = 1 and it is negatively skewed, left-tailed. There is '''one inflection point''', located to the left of the mode: ::<math>x = \frac{\alpha -1 -\sqrt{\frac{(\alpha-1)(\beta-1)}{\alpha+\beta-3}}}{\alpha+\beta-2}</math> There are no inflection points in the remaining (symmetric and skewed) regions: U-shaped: (''α'', ''β'' < 1) upside-down-U-shaped: (1 < ''α'' < 2, 1 < ''β'' < 2), reverse-J-shaped (''α'' < 1, ''β'' > 2) or J-shaped: (''α'' > 2, ''β'' < 1) The accompanying plots show the inflection point locations (shown vertically, ranging from 0 to 1) versus ''α'' and ''β'' (the horizontal axes ranging from 0 to 5). There are large cuts at surfaces intersecting the lines ''α'' = 1, ''β'' = 1, ''α'' = 2, and ''β'' = 2 because at these values the beta distribution change from 2 modes, to 1 mode to no mode. ====Shapes==== [[File:PDF for symmetric beta distribution vs. x and alpha=beta from 0 to 30 - J. Rodal.jpg|thumb|PDF for symmetric beta distribution vs. ''x'' and ''α'' = ''β'' from 0 to 30]] [[File:PDF for symmetric beta distribution vs. x and alpha=beta from 0 to 2 - J. Rodal.jpg|thumb|PDF for symmetric beta distribution vs. x and ''α'' = ''β'' from 0 to 2]] [[File:PDF for skewed beta distribution vs. x and beta= 2.5 alpha from 0 to 9 - J. Rodal.jpg|thumb|PDF for skewed beta distribution vs. ''x'' and ''β'' = 2.5''α'' from 0 to 9]] [[File:PDF for skewed beta distribution vs. x and beta= 5.5 alpha from 0 to 9 - J. Rodal.jpg|thumb|PDF for skewed beta distribution vs. x and ''β'' = 5.5''α'' from 0 to 9]] [[File:PDF for skewed beta distribution vs. x and beta= 8 alpha from 0 to 10 - J. Rodal.jpg|thumb|PDF for skewed beta distribution vs. x and ''β'' = 8''α'' from 0 to 10]] The beta density function can take a wide variety of different shapes depending on the values of the two parameters ''α'' and ''β''. The ability of the beta distribution to take this great diversity of shapes (using only two parameters) is partly responsible for finding wide application for modeling actual measurements: =====Symmetric (''α'' = ''β'')===== * the density function is [[symmetry|symmetric]] about 1/2 (blue & teal plots). * median = mean = 1/2. *skewness = 0. *variance = 1/(4(2''α'' + 1)) *'''''α'' = ''β'' < 1''' **U-shaped (blue plot). **bimodal: left mode = 0, right mode =1, anti-mode = 1/2 **1/12 < var(''X'') < 1/4<ref name=JKB/> **−2 < excess kurtosis(''X'') < −6/5 ** ''α'' = ''β'' = 1/2 is the [[arcsine distribution]] *** var(''X'') = 1/8 ***excess kurtosis(''X'') = −3/2 ***CF = Rinc (t) <ref>{{Cite book|last1=Buchanan|first1=K.|last2=Rockway|first2=J.|last3=Sternberg|first3=O.|last4=Mai|first4=N. N.|title=2016 IEEE Radar Conference (RadarConf) |chapter=Sum-difference beamforming for radar applications using circularly tapered random arrays |date=May 2016|pages=1–5|doi=10.1109/RADAR.2016.7485289|isbn=978-1-5090-0863-6|s2cid=32525626|chapter-url=https://zenodo.org/record/1279364}}</ref> ** ''α'' = ''β'' → 0 is a 2-point [[Bernoulli distribution]] with equal probability 1/2 at each [[Dirac delta function]] end ''x'' = 0 and ''x'' = 1 and zero probability everywhere else. A coin toss: one face of the coin being ''x'' = 0 and the other face being ''x'' = 1. *** <math> \lim_{\alpha = \beta \to 0} \operatorname{var}(X) = \tfrac{1}{4} </math> *** <math> \lim_{\alpha = \beta \to 0} \operatorname{excess \ kurtosis}(X) = - 2</math> a lower value than this is impossible for any distribution to reach. *** The [[information entropy|differential entropy]] approaches a [[Maxima and minima|minimum]] value of −∞ *'''α = β = 1''' **the [[uniform distribution (continuous)|uniform [0, 1] distribution]] **no mode **var(''X'') = 1/12 **excess kurtosis(''X'') = −6/5 **The (negative anywhere else) [[information entropy|differential entropy]] reaches its [[Maxima and minima|maximum]] value of zero **CF = Sinc (t) *'''''α'' = ''β'' > 1''' **symmetric [[unimodal]] ** mode = 1/2. **0 < var(''X'') < 1/12<ref name=JKB/> **−6/5 < excess kurtosis(''X'') < 0 **''α'' = ''β'' = 3/2 is a semi-elliptic [0, 1] distribution, see: [[Wigner semicircle distribution]]<ref>{{Cite book|last1=Buchanan|first1=K.|last2=Flores|first2=C.|last3=Wheeland|first3=S.|last4=Jensen|first4=J.|last5=Grayson|first5=D.|last6=Huff|first6=G.|title=2017 IEEE Radar Conference (RadarConf) |chapter=Transmit beamforming for radar applications using circularly tapered random arrays |date=May 2017|pages=0112–0117|doi=10.1109/RADAR.2017.7944181|isbn=978-1-4673-8823-8|s2cid=38429370}}</ref> ***var(''X'') = 1/16. ***excess kurtosis(''X'') = −1 ***CF = 2 Jinc (t) **''α'' = ''β'' = 2 is the parabolic [0, 1] distribution ***var(''X'') = 1/20 ***excess kurtosis(''X'') = −6/7 ***CF = 3 Tinc (t) <ref>{{Cite web|last=Ryan|first=Buchanan, Kristopher|date=2014-05-29|title=Theory and Applications of Aperiodic (Random) Phased Arrays|url=http://oaktrust.library.tamu.edu/handle/1969.1/157918|language=en}}</ref> **''α'' = ''β'' > 2 is bell-shaped, with [[inflection point]]s located to either side of the mode ***0 < var(''X'') < 1/20 ***−6/7 < excess kurtosis(''X'') < 0 **''α'' = ''β'' → ∞ is a 1-point [[Degenerate distribution]] with a [[Dirac delta function]] spike at the midpoint ''x'' = 1/2 with probability 1, and zero probability everywhere else. There is 100% probability (absolute certainty) concentrated at the single point ''x'' = 1/2. ***<math> \lim_{\alpha = \beta \to \infty} \operatorname{var}(X) = 0 </math> ***<math> \lim_{\alpha = \beta \to \infty} \operatorname{excess \ kurtosis}(X) = 0</math> ***The [[information entropy|differential entropy]] approaches a [[Maxima and minima|minimum]] value of −∞ =====Skewed (''α'' ≠ ''β'')===== The density function is [[Skewness|skewed]]. An interchange of parameter values yields the [[mirror image]] (the reverse) of the initial curve, some more specific cases: *'''''α'' < 1, ''β'' < 1''' ** U-shaped ** Positive skew for ''α'' < ''β'', negative skew for ''α'' > ''β''. ** bimodal: left mode = 0, right mode = 1, anti-mode = <math>\tfrac{\alpha-1}{\alpha + \beta-2} </math> ** 0 < median < 1. ** 0 < var(''X'') < 1/4 *'''''α'' > 1, ''β'' > 1''' ** [[unimodal]] (magenta & cyan plots), **Positive skew for ''α'' < ''β'', negative skew for ''α'' > ''β''. **<math>\text{mode}= \tfrac{\alpha-1}{\alpha + \beta-2} </math> ** 0 < median < 1 ** 0 < var(''X'') < 1/12 *'''''α'' < 1, ''β'' ≥ 1''' **reverse J-shaped with a right tail, **positively skewed, **strictly decreasing, [[convex function|convex]] ** mode = 0 ** 0 < median < 1/2. ** <math>0 < \operatorname{var}(X) < \tfrac{-11+5 \sqrt{5}}{2}, </math> (maximum variance occurs for <math>\alpha=\tfrac{-1+\sqrt{5}}{2}, \beta=1</math>, or ''α'' = '''Φ''' the [[Golden ratio|golden ratio conjugate]]) *'''''α'' ≥ 1, ''β'' < 1''' **J-shaped with a left tail, **negatively skewed, **strictly increasing, [[convex function|convex]] ** mode = 1 ** 1/2 < median < 1 ** <math>0 < \operatorname{var}(X) < \tfrac{-11+5 \sqrt{5}}{2},</math> (maximum variance occurs for <math>\alpha=1, \beta=\tfrac{-1+\sqrt{5}}{2}</math>, or ''β'' = '''Φ''' the [[Golden ratio|golden ratio conjugate]]) *'''''α'' = 1, ''β'' > 1''' **positively skewed, **strictly decreasing (red plot), **a reversed (mirror-image) power function [0,1] distribution ** mean = 1 / (''β'' + 1) ** median = 1 - 1/2<sup>1/''β''</sup> ** mode = 0 **α = 1, 1 < β < 2 ***[[concave function|concave]] *** <math>1-\tfrac{1}{\sqrt{2}}< \text{median} < \tfrac{1}{2}</math> *** 1/18 < var(''X'') < 1/12. **α = 1, β = 2 ***a straight line with slope −2, the right-[[triangular distribution]] with right angle at the left end, at ''x'' = 0 *** <math>\text{median}=1-\tfrac {1}{\sqrt{2}}</math> *** var(''X'') = 1/18 **α = 1, β > 2 ***reverse J-shaped with a right tail, ***[[convex function|convex]] *** <math>0 < \text{median} < 1-\tfrac{1}{\sqrt{2}}</math> *** 0 < var(''X'') < 1/18 *'''α > 1, β = 1''' **negatively skewed, **strictly increasing (green plot), **the power function [0, 1] distribution<ref name="Handbook of Beta Distribution" /> ** mean = α / (α + 1) ** median = 1/2<sup>1/α </sup> ** mode = 1 **2 > α > 1, β = 1 ***[[concave function|concave]] *** <math>\tfrac{1}{2} < \text{median} < \tfrac{1}{\sqrt{2}}</math> *** 1/18 < var(''X'') < 1/12 ** α = 2, β = 1 ***a straight line with slope +2, the right-[[triangular distribution]] with right angle at the right end, at ''x'' = 1 *** <math>\text{median}=\tfrac {1}{\sqrt{2}}</math> *** var(''X'') = 1/18 **α > 2, β = 1 ***J-shaped with a left tail, [[convex function|convex]] ***<math>\tfrac{1}{\sqrt{2}} < \text{median} < 1</math> *** 0 < var(''X'') < 1/18
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