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A continuous probability distribution concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b0159501.png" /> with density
+
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b0159502.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b0159503.png" /> are non-negative and the normalizing factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b0159504.png" /> is Euler's beta-function
+
A continuous probability distribution concentrated on  $  (0, 1) $
 +
with density
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b0159505.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
\beta _ {m, n }  (x)  = \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b0159506.png" /> is the [[Gamma-function|gamma-function]]. The distribution function is expressed as the incomplete beta-function
+
\frac{1}{B (m, n) }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b0159507.png" /></td> </tr></table>
+
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
 
(this function has been tabulated, see [[#References|[1]]], [[#References|[2]]]). The moments of the beta-distribution are given by the formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b0159508.png" /></td> </tr></table>
+
$$
 +
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)  = \
  
In particular, the mathematical expectation and the variance are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b0159509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595010.png" />, respectively. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595012.png" />, the density curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595013.png" /> has a single mode at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595014.png" /> and vanishes at the ends of the interval. If either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595015.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595016.png" />, one ordinate at the end of the graph becomes infinite, and if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595018.png" />, both ordinates at the ends of the interval are infinite and the curve is U-shaped. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595020.png" /> the beta-distribution reduces to the [[Uniform distribution|uniform distribution]] on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595021.png" />. Another special case of the beta-distribution is the so-called [[Arcsine distribution|arcsine distribution]]:
+
\frac{1}{B (m, n) }
 +
\cdot
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595022.png" /></td> </tr></table>
+
\frac{t ^ {m - 1 } }{(1 + t) ^ {m + n - 2 } }
 +
,\ \
 +
0 < t < \infty .
 +
$$
  
If one substitutes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595023.png" /> in (1), then one obtains a distribution having the density
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$
 +
F _ {m, n }  = \
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595026.png" /> are independent and have gamma-distributions (cf. [[Gamma-distribution|Gamma-distribution]]) with respective parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595028.png" />, then the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595029.png" /> will have a beta-distribution with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595030.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595031.png" />-relationship
+
\frac{n \chi _ {m}  ^ {2} }{m \chi _ {n} ^ {2} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595032.png" /></td> </tr></table>
+
$$
  
(the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595033.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595034.png" />-distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595035.png" /> degrees of freedom) is expressed by the formula
+
(the random variable $  \chi _ {k}  ^ {2} $
 +
has a $  \chi  ^ {2} $-
 +
distribution with $  k $
 +
degrees of freedom) is expressed by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595036.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} (F _ {m, n }  < x)  = \
 +
B _ {m/2, n/2 }
 +
\left (
 +
\frac{mx}{n + mx }
 +
\right )
 +
$$
  
(the values of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595037.png" />-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
+
(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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015950/b01595038.png" /></td> </tr></table>
+
$$
 +
B _ {n - m, m + 1 }
 +
(1 - p)  = \
 +
\sum _ {k = 0 } ^ { m }
 +
\left (
 +
\begin{array}{c}
 +
n \\
 +
k
 +
\end{array}
 +
\
 +
\right )
 +
p  ^ {k}
 +
(1 - p) ^ {n - k } .
 +
$$
  
 
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]].
 
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]].

Latest revision as of 10:58, 29 May 2020


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)
How to Cite This Entry:
Beta-distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Beta-distribution&oldid=16107
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article