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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|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 [[#References|[1]]], [[#References|[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|uniform distribution]] on the interval  $  (0, 1) $. 
Another special case of the beta-distribution is the so-called [[Arcsine distribution|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|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|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|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|Jacobi polynomials]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Pearson,   "Tables of the incomplete beta-function" , Cambridge Univ. Press  (1932)</TD></TR></table>