Editing Beta distribution (section)

====Mode====
The [[Mode (statistics)|mode]] of a beta distributed [[random variable]] ''X'' with ''α'', ''β'' > 1 is the most likely value of the distribution (corresponding to the peak in the PDF), and is given by the following expression:<ref name=JKB>{{cite book|last1=Johnson|first1= Norman L. |first2= Samuel|last2= Kotz |first3= N. |last3= Balakrishnan| year=1995 |title=Continuous Univariate Distributions Vol. 2 |edition=2nd |publisher= Wiley |isbn= 978-0-471-58494-0 |chapter= Chapter 25: Beta Distributions}}</ref>

:<math>\frac{\alpha - 1} {\alpha + \beta - 2} .</math>

When both parameters are less than one (''α'', ''β'' < 1), this is the anti-mode: the lowest point of the probability density curve.<ref name=Wadsworth>{{cite book|last=Wadsworth |first=George P. and Joseph Bryan |title=Introduction to Probability and Random Variables|url=https://archive.org/details/introductiontopr0000wads |url-access=registration |year=1960|publisher=McGraw-Hill}}</ref>

Letting ''α'' = ''β'', the expression for the mode simplifies to 1/2, showing that for ''α'' = ''β'' > 1 the mode (resp. anti-mode when {{nowrap|''α'', ''β'' < 1}}), is at the center of the distribution: it is symmetric in those cases.  See [[Beta distribution#Shapes|Shapes]] section in this article for a full list of mode cases, for arbitrary values of ''α'' and ''β''. For several of these cases, the maximum value of the density function occurs at one or both ends.  In some cases the (maximum) value of the density function occurring at the end is finite. For example, in the case of ''α'' = 2, ''β'' = 1 (or ''α'' = 1, ''β'' = 2), the density function becomes a [[Triangular distribution|right-triangle distribution]] which is finite at both ends. In several other cases there is a [[Mathematical singularity|singularity]] at one end, where the value of the density function approaches infinity. For example, in the case ''α'' = ''β'' = 1/2, the beta distribution simplifies to become the [[arcsine distribution]].  There is debate among mathematicians about some of these cases and whether the ends (''x'' = 0, and ''x'' = 1) can be called ''modes'' or not.<ref name="Handbook of Beta Distribution" /><ref name="Mathematical Statistics with MATHEMATICA">{{cite book |last1=Rose |first1=Colin |last2=Smith |first2=Murray D. |title=Mathematical Statistics with MATHEMATICA |year=2002 |publisher=Springer |isbn=978-0387952345}}</ref>
[[File:Mode Beta Distribution for alpha and beta from 1 to 5 - J. Rodal.jpg|325px|thumb|Mode for beta distribution for 1 ≤ ''α'' ≤ 5 and 1 ≤ β ≤ 5]]

* Whether the ends are part of the [[Domain of a function|domain]] of the density function
* Whether a [[Mathematical singularity|singularity]] can ever be called a ''mode''
* Whether cases with two maxima should be called ''bimodal''