Non-Normal Distributions - Beta Distribution

The Beta distribution has the probability density function:

f(x) = Γ(ν+ω)/(Γ(ν)Γ(ω)) * xν-1 * (1-x)ω-1

0 < x < 1, ν > 0, ω > 0

where

Γ (Gamma) is the Gamma function
ν, ω are parameters

(Note that the shape of the Beta distribution depends on the values of the parameters as shown in the animation below; for more information, see Hahn and Shapiro, 1994.)

Threshold (location) and sigma

The standardized Beta distribution has a valid range from 0 to 1. However, you may specify a lower threshold (location) parameter and a scale parameter (Sigma); the program will then fit the Beta distribution to the standardized variate computed as:

(x-location)/Sigma

Note: the location parameter must be less than the smallest observed value, and the location+Sigma must be greater than the largest observed value.

Applications

Because the Beta distribution is bounded on both sides, it is often used for representing processes with natural lower and upper limits. For examples, refer to Hahn and Shapiro (1967).

Parameter estimation

Statistica computes maximum likelihood parameter estimates for the Beta distribution (see Evans, Hastings, & Peacock, 1993, for details).