# Beta-distribution

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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 , ). 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.

How to Cite This Entry:
Beta-distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Beta-distribution&oldid=46045
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article