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
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where is the gamma-function. The distribution function is expressed as the incomplete beta-function
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(this function has been tabulated, see [1], [2]). The moments of the beta-distribution are given by the formulas
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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:
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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
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(the random variable has a
-distribution with
degrees of freedom) is expressed by the formula
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(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
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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) |
Beta-distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Beta-distribution&oldid=46045