A general type of statistical distribution which is related to the gamma distribution.
Beta distributions have two free parameters, which are labeled according to one of
two notational conventions. The usual definition calls these 
and 
, and the other uses 
and 
(Beyer 1987, p. 534). The beta distribution
is used as a prior distribution for binomial
proportions in Bayesian analysis (Evans et
al. 2000, p. 34). The above plots are for various values of 
with 
and 
ranging from 0.25 to 3.00.
The domain is ![[0,1]](/images/equations/BetaDistribution/Inline8.svg)
,
and the probability function 
and distribution function 
are given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is the beta function, 
is the regularized
beta function, and 
. The beta distribution is implemented in the
Wolfram Language as BetaDistribution[alpha,
beta].
The distribution is normalized since
 |
(4)
|
The characteristic function is
 |  |  |
(5)
|
 |  |  |
(6)
|
where 
is a confluent hypergeometric
function of the first kind.
The raw moments are given by
 |  |  |
(7)
|
 |  |  |
(8)
|
(Papoulis 1984, p. 147), and the central moments by
 |
(9)
|
where 
is a hypergeometric function.
The mean, variance, skewness, and kurtosis excess are therefore given by
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  | ![(6[alpha^3+alpha^2(1-2beta)+beta^2(1+beta)-2alphabeta(2+beta)])/(alphabeta(alpha+beta+2)(alpha+beta+3)).](/images/equations/BetaDistribution/Inline48.svg) |
(13)
|
The mode of a variate distributed as 
is
 |
(14)
|
Independent Random Variables whose Product
Follows the Beta Distribution." Colloq. Math. IX Fasc. 2, 325-332,
1962.Krysicki, W. "On Some New Properties of the Beta Distribution."
Stat. Prob. Let. 42, 131-137, 1999.Papoulis, A.