Beta Distribution - Properties - Geometry of The Probability Distribution Function - Inflection Points

Inflection Points

For certain values of the shape parameters α and β, the probability distribution function has inflection points, at which the curvature changes sign. The position of these inflection points can be useful as a measure of the dispersion or spread of the distribution.

Defining the following quantity:

Points of inflection occur, depending on the value of the shape parameters α and β,as follows:

• The distribution is bell-shaped (symmetric for α = β and skewed otherwise), with two inflection points, equidistant from the mode as follows:

left of the mode at:

right of the mode at:

• The distribution is unimodal, positively skewed, right-tailed, with one inflection point, located to the right of the mode, as follows:

• The distribution is unimodal, negatively skewed, left-tailed, with one inflection point, located to the left of the mode, as follows:

• The distribution is unimodal, positively skewed, right-tailed, with one inflection point, located to the right of the mode, as follows:

• 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, at:

• The distribution is unimodal negatively skewed, left-tailed, with one inflection point, located to the left of the mode, as follows:

• 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, at:

There are no inflection points in the remaining (symmetric and skewed) regions: U-shaped: upside-down-U-shaped:, reverse-J-shaped or J-shaped:

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.