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Beta Binomial Distribution

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A variable with a beta binomial distribution is distributed as a binomial distribution with parameter p, where p is distribution with a beta distribution with parameters alpha and beta. For n trials, it has probability density function

P(x)=(B(x+alpha,n-x+beta)(n; x))/(B(alpha,beta)),
(1)

where B(a,b) is a beta function and (n; k) is a binomial coefficient, and distribution function

D(x)=1-(nB(b+n-x-1,a+x+1)Gamma(n)F_n(a,b;x))/(B(a,b)B(n-x,x+2)Gamma(n+2)),
(2)

where Gamma(n) is a gamma function and

F_n(a,b;x) =_3F_2(1,a+x+1,-n+x+1;x+2,-b-n+x+2;1)
(3)

is a generalized hypergeometric function.

It is implemented as BetaBinomialDistribution[alpha, beta, n].

The first few raw moments are

mu_1^'=(nalpha)/(alpha+beta)
(4)
mu_2^'=(nalpha[n(1+alpha)+beta])/((alpha+beta)(1+alpha+beta))
(5)
mu_3^'=(nalpha[n^2(1+alpha)(2+alpha)+3n(1+alpha)beta+beta(beta-alpha)])/((alpha+beta)(1+alpha+beta)(2+alpha+beta)),
(6)

giving the mean and variance as

mu=(nalpha)/(alpha+beta)
(7)
sigma^2=(nalphabeta(n+alpha+beta))/((alpha+beta)^2(1+alpha+beta)).
(8)

See also

Beta Distribution, Binomial Distribution

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Cite this as:

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

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