A general sort of distribution which is said to the gamma distribution. Beta distributions have two free parameters, which are labeled consistent with one among two notational conventions. the standard definition calls these alpha and beta, and therefore the other uses beta^’=beta-1 and alpha^’=alpha-1 (Beyer 1987, p. 534). The beta distribution is employed as a previous 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 starting from 0.25 to 3.00.

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

The distribution is normalized since

The characteristic function is

where _1F_1(a;b;z) may be a confluent hypergeometric function of the primary kind.

The raw moments are given by

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

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

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

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This mode of a variate distributed as is