Editing Beta distribution (section)

====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.