Beta-distribution

A continuous probability distribution concentrated on with density

(1)

where the parameters are non-negative and the normalizing factor is Euler's beta-function

where is the gamma-function. The distribution function is expressed as the incomplete beta-function

(this function has been tabulated, see [1], [2]). The moments of the beta-distribution are given by the formulas

In particular, the mathematical expectation and the variance are and , respectively. If and , the density curve has a single mode at the point and vanishes at the ends of the interval. If either or , one ordinate at the end of the graph becomes infinite, and if both and , both ordinates at the ends of the interval are infinite and the curve is U-shaped. If and the beta-distribution reduces to the uniform distribution on the interval . Another special case of the beta-distribution is the so-called arcsine distribution:

If one substitutes in (1), then one obtains a distribution having the density

(2)

This distribution is called a beta-distribution of the second kind, as distinct from the beta-distribution (1). The distributions (1) and (2) correspond to "type I" and "type VI" distributions in the system of Pearson curves. An important case of generation of a beta-distribution is the following: If and are independent and have gamma-distributions (cf. Gamma-distribution) with respective parameters and , then the random variable will have a beta-distribution with density . This fact to a large extent explains the role played by beta-distributions in various applications, in particular in mathematical statistics: The distributions of several important statistics are reducible to beta-distributions. For instance, the distribution function of the -relationship

(the random variable has a -distribution with degrees of freedom) is expressed by the formula

(the values of the -distribution are usually calculated with the aid of tables of beta-functions). The beta-distribution function also allows one to compute the values of the binomial distribution functions, in view of the relationship

Beta-distributions are used in fields other than mathematical statistics; thus, the density of the beta-distribution is the weight function for the system of orthogonal Jacobi polynomials.

References

[1]	L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)
[2]	K. Pearson, "Tables of the incomplete beta-function" , Cambridge Univ. Press (1932)