The standard form of the Beta distribution is a two parameter distribution whose values extend over a finite domain, [0,1]. A more complex version is also sometimes cited, in which the domain of the function is over the range [a,b], but it is generally possible to transform sample data to lie within the range [0,1] and apply the standard version of the distribution function.
The probability density function is defined in terms of the Gamma function, with two shape parameters, α≥1 and β≥1, as:
where B(α,β) is the Beta function. The cumulative distribution function is denoted as:
and is known as the incomplete beta function or incomplete beta function ratio. If α=1 we have the standard power function density:
and if α=β=1 we have the uniform distribution, f(x)=1, over the interval [0.1]. The cases α=2 β=1 and α=1 β=2 both yield straight line graphs, since one or other of the terms in x disappears. Plots of the standard distribution for four other sample values of the two parameters are shown below:
Apart from the obvious usefulness of this distribution as a convenient model for many different datasets, it has a number of applications reflecting ways in which it may be generated. One such example relates to order statistics. Suppose that y1, y2, ..., yn are independent random variables drawn from a Uniform distribution over the interval [0,1]. If these are re-arranged in order of size so that we have the new set of ordered values: Y1, Y2, ..., Yn or order statistics. The kth order statistic in this instance has a Beta distribution: B(k,n-k+1). A second example (empirically derived by the current author) is the distribution of distances within bounded convex figures. If a large sample of randomly selected pairs of points are taken within any convex polygon, their lengths computed and then the set of distances standardized by dividing each by the largest value in the sample, the distribution of such distances will be closely approximated by a Beta distribution. A final, important application of the Beta is in connection with Bayesian analysis, where the Beta distribution provides the conjugate prior for a number of discrete distributions.
Key measures for the distribution are provided in the table below:
Wikipedia: Beta distribution: https://en.wikipedia.org/wiki/Beta_distribution
Wikipedia: Conjugate Priors: https://en.wikipedia.org/wiki/Conjugate_prior_distribution