Beta-distribution

A continuous probability distribution concentrated on $ (0, 1) $ with density

$$ \tag{1 } \beta _ {m, n } (x) = \ \frac{1}{B (m, n) } x ^ {m - 1 } (1 - x) ^ {n - 1 } , $$

where the parameters $ m, n $ are non-negative and the normalizing factor $ B(m, n) $ is Euler's beta-function

$$ B (m, n) = \ \int\limits _ { 0 } ^ { 1 } x ^ {m - 1 } (1 - x) ^ {n - 1 } \ dx = \frac{\Gamma (m) \Gamma (n) }{\Gamma (m + n) } , $$

where $ \Gamma (n) $ is the gamma-function. The distribution function is expressed as the incomplete beta-function

$$ B _ {m, n } (x) = \ \frac{1}{B (m, n) } \int\limits _ { 0 } ^ { x } y ^ {m - 1 } (1 - y) ^ {n - 1 } dy,\ \ 0 < x < 1 $$

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

$$ m _ {k} = \ \frac{B (m + k, n) }{B (m, n) } ,\ \ k = 1, 2 , . . . . $$

In particular, the mathematical expectation and the variance are $ m/(m + n) $ and $ mn/ \{ (m + n) ^ {2} (m + n + 1) \} $, respectively. If $ m > 1 $ and $ n > 1 $, the density curve $ \beta _ {m,n} (x) $ has a single mode at the point $ x = (m - 1)/(m + n - 2) $ and vanishes at the ends of the interval. If either $ m < 1 $ or $ n < 1 $, one ordinate at the end of the graph becomes infinite, and if both $ m < 1 $ and $ n < 1 $, both ordinates at the ends of the interval are infinite and the curve is U-shaped. If $ m = 1 $ and $ n = 1 $ the beta-distribution reduces to the uniform distribution on the interval $ (0, 1) $. Another special case of the beta-distribution is the so-called arcsine distribution:

$$ \beta _ {1/2, 1/2 } (x) = \ \frac{1}{\pi \sqrt {x (1 - x) } } . $$

If one substitutes $ x = 1/(1 + t) $ in (1), then one obtains a distribution having the density

$$ \tag{2 } \beta _ {m,n} ^ \prime (t) = \ \frac{1}{B (m, n) } \cdot \frac{t ^ {m - 1 } }{(1 + t) ^ {m + n - 2 } } ,\ \ 0 < t < \infty . $$

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 $ X _ {1} $ and $ X _ {2} $ are independent and have gamma-distributions (cf. Gamma-distribution) with respective parameters $ m $ and $ n $, then the random variable $ X _ {1} / (X _ {1} + X _ {2} ) $ will have a beta-distribution with density $ \beta _ {m,n} (x) $. 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 $ F $- relationship

$$ F _ {m, n } = \ \frac{n \chi _ {m} ^ {2} }{m \chi _ {n} ^ {2} } $$

(the random variable $ \chi _ {k} ^ {2} $ has a $ \chi ^ {2} $- distribution with $ k $ degrees of freedom) is expressed by the formula

$$ {\mathsf P} (F _ {m, n } < x) = \ B _ {m/2, n/2 } \left ( \frac{mx}{n + mx } \right ) $$

(the values of the $ F $- 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

$$ B _ {n - m, m + 1 } (1 - p) = \ \sum _ {k = 0 } ^ { m } \left ( \begin{array}{c} n \\ k \end{array} \ \right ) p ^ {k} (1 - p) ^ {n - k } . $$

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)