From a theoretical perspective, the Beta distribution is intimately related to the Binomial distribution. In this context it represents the distribution of the uncertainty of the probability parameter of a Binomial process based on a certain number of observations. As a distribution of probability, its range is clearly between zero and one, and as it is related to noting successful outcomes within a number of trials, it is a two-parameter distribution. The BetaGeneral is directly derived from the Beta distribution by scaling the range of the Beta distribution with the use of minimum and maximum values, and is hence a four parameter distribution (min, max, alpha1, alpha2).
Special cases of the BetaGeneral distribution are however perhaps most frequently used as a way to build models, where a pragmatic modelling approach is desired. For example, the PERT distribution is a special case of the BetaGeneral distribution, using the three parameters min, most likely, and max. If the parameters alpha1, alpha2 of the BetaGeneral are derived from those of the PERT distribution (e.g. by setting alpha1 equal to 6*(µ-min)/(max-min), where µ (the mean of the PERT) is equal to (min+4*ML+max)/6, and alpha1 equal to 6*(max-µ)/(max-min). The BetaSubjective distribution is another variation, and requires the four parameters min, most-likely, mean, max. In a sense it is quite an unusual distribution, having both the mode and the mean as a parameter, and in some contexts can therefore be quite useful when one has knowledge of both of these and requires a distribution to capture that.
Dr. Michael Rees
Director of Training and Consulting