Editing Beta distribution (section)

=====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