Editing Beta distribution (section)

=====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 −∞