## Beta Distribution

A general type of statistical Distribution which is related to the Gamma Distribution. Beta distributions have two free parameters, which are labeled according to one of two notational conventions. The usual definition calls these and , and the other uses and (Beyer 1987, p. 534). The above plots are for [solid], (1, 2) [dotted], and (2, 3) [dashed]. The probability function and Distribution Function are given by

 (1) (2)

where is the Beta Function, is the Regularized Beta Function, and where , . The distribution is normalized since
 (3) (4)

The Characteristic Function is
 (5)

The Moments are given by
 (6)

The Mean is
 (7)

and the Variance, Skewness, and Kurtosis are
 (8) (9) (10)

The Mode of a variate distributed as is
 (11)

In normal'' form, the distribution is written

 (12)

and the Mean, Variance, Skewness, and Kurtosis are
 (13) (14) (15) (16)