Beta Distribution

see section: Fisher information matrix

In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval parametrized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. In population genetics it has been employed for a statistical description of the allele frequencies in the components of a sub-divided population. It has been utilized in PERT, 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. In acoustic analysis, the kurtosis of the beta distribution has been reported to be a good indicator of gear damage. Sunshine data for application to solar renewable energy utilization was modeled with a beta distribution. It has 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. It was selected to determine well-log shale parameters, to describe the proportions of the mineralogical components existing in a certain stratigraphic interval. Heterogeneity in the probability of HIV transmission in heterosexual contact has been modeled as a random variable in a beta distribution, and parameters estimated by maximum-likelihood. In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, 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 and proportions. 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).

The usual formulation of the beta distribution is also known as the beta distribution of the first kind, whereas beta distribution of the second kind is an alternative name for the beta prime distribution.