Beta Distribution

Abstract

In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parametrized by two positive shape parameters, denoted by α and β. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length. It has been used in population genetics for a statistical description of the allele frequencies in the components of a sub-divided population[1] . It has also been used extensively in PERT[2], critical path method (CPM) and other project management / control systems to describe the statistical distributions of the time to completion and the cost of a task. It has also been applied in acoustic analysis to assess damage to gears, as the kurtosis of the beta distribution has been reported as a good indicator of the condition of gears.[3] It has also been used to model sunshine data for application to solar renewable energy utilization.[4] It has also been used for parametrizing variability of soil properties at the regional level for crop yield estimation, modeling crop response over the area of the association.[5] It has also been used to determine well-log shale parameters, to describe the proportions of the mineralogical components existing in a certain stratigraphic interval.[6] It is used extensively in Bayesian inference, since beta distributions provide a family of conjugate prior distributions for binomial and geometric distributions. For example, the beta distribution can be used in Bayesian analysis to describe initial knowledge concerning probability of success such as the probability that a space vehicle will successfully complete a specified mission. The beta distribution is a suitable model for the random behavior of percentages. It can be suited to the statistical modelling of proportions in applications where values of proportions equal to 0 or 1 do not occur. One theoretical case where the beta distribution arises is as the distribution of the ratio formed by one random variable having a Gamma distribution divided by the sum of it and another independent random variable also having a Gamma distribution with the same scale parameter (but possibly different shape parameter).