Beta-binomial distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ betabinomial(a,b,n) is used to indicate that the random variable X has the beta-binomial distribution with parameters a, b, and n where a,b > 0 and n is a positive integer. A beta-binomial random variable X with parameters a, b, and n has probability mass function

f(x) = \frac{Γ(x + a) Γ(n − x + b) Γ(a + b) Γ(n + 2)}{(n + 1) Γ(a + b + n) Γ(a)Γ(b)Γ(x + 1) Γ(n − x + 1)} x = 0,1,...,n.

A beta-binomial random variable is a binomial random variable with a random parameter p which has the beta distribution with parameters a and b. The probability mass function for n = 20 and three different parameter settings is illustrated below.

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x f(x)

a = 0.7 b = 2 a = 2 a = 6 b = 4

The cumulative distribution function, survivor function, hazard function, cumulative hazard function, inverse distribution function, moment generating function, and characteristic function on the support of X are mathematically intractable. The population mean, variance, and skewness of X are

E[X] = \frac{na}{a + b} V [X] = \frac{nab(a + b + n)}{(a + b)^{2}(a + b + 1)}

E [( \frac{X − μ}{σ} ) ^{3}] = \frac{(a + b + 2n)(b − a)}{(a + b + 2)} \sqrt{\frac{1 + a + b}{nab(n + a + b)}} .

The population kurtosis of X is mathematically intractable.