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Beta Distribution

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BetaDistribution

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 alpha and beta, and the other uses beta^'=beta-1 and alpha^'=alpha-1 (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 (alpha,beta) with alpha=1 and beta ranging from 0.25 to 3.00.

The domain is [0,1], and the probability function P(x) and distribution function D(x) are given by

P(x)=((1-x)^(beta-1)x^(alpha-1))/(B(alpha,beta))
(1)
=(Gamma(alpha+beta))/(Gamma(alpha)Gamma(beta))(1-x)^(beta-1)x^(alpha-1)
(2)
D(x)=I(x;a,b),
(3)

where B(a,b) is the beta function, I(x;a,b) is the regularized beta function, and alpha,beta>0. The beta distribution is implemented in the Wolfram Language as BetaDistribution[alpha, beta].

The distribution is normalized since

int_0^1P(x)dx=1.
(4)

The characteristic function is

phi(t)=int_0^1(x^(a-1)(1-x)^(b-1))/(beta(a,b))e^(-2piixt)dx
(5)
=_1F_1(a;a+b;it),
(6)

where _1F_1(a;b;z) is a confluent hypergeometric function of the first kind.

The raw moments are given by

mu_r^'=int_0^1P(x)x^rdx
(7)
=(Gamma(alpha+beta)Gamma(alpha+r))/(Gamma(alpha+beta+r)Gamma(alpha))
(8)

(Papoulis 1984, p. 147), and the central moments by

mu_r=(-alpha/(alpha+beta))^r_2F_1(alpha,-r;alpha+beta;(alpha+beta)/alpha),
(9)

where _2F_1(a,b;c;x) is a hypergeometric function.

The mean, variance, skewness, and kurtosis excess are therefore given by

mu=alpha/(alpha+beta)
(10)
sigma^2=(alphabeta)/((alpha+beta)^2(alpha+beta+1))
(11)
gamma_1=(2(beta-alpha)sqrt(1+alpha+beta))/(sqrt(alphabeta)(2+alpha+beta))
(12)
gamma_2=(6[alpha^3+alpha^2(1-2beta)+beta^2(1+beta)-2alphabeta(2+beta)])/(alphabeta(alpha+beta+2)(alpha+beta+3)).
(13)

The mode of a variate distributed as beta(alpha,beta) is

x^^=(alpha-1)/(alpha+beta-2).
(14)

See also

Gamma Distribution

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References

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 n 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.

Referenced on Wolfram|Alpha

Beta Distribution

Cite this as:

Weisstein, Eric W. "Beta Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BetaDistribution.html

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