Beta Distribution
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]](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/BetaDistribution/Inline8.gif)
, 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 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)).](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/BetaDistribution/Inline48.gif) |
(13)
|
The mode of a variate distributed as 
is
 |
(14)
|
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 944-945, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 534-535, 1987.
Evans, M.; Hastings, N.; and Peacock, B. "Beta Distribution." Ch. 5 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 34-42, 2000.
Jambunathan, M. V. "Some Properties of Beta and Gamma Distributions." Ann. Math. Stat. 25, 401-405, 1954.
Kolarski, I. "On Groups of 
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. The Fourier Integral and Its Applications. New York: McGraw-Hill, 1962.
Weisstein, Eric W. "Beta Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BetaDistribution.html
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