Beta-binomial compound distribution (exact PMF and moments)
BetaBinomialDistribution[α, β, n]
Mean[BetaBinomialDistribution[α, β, n]]
→ (n*α)/(α + β)PDF[BetaBinomialDistribution[α, β, n], x]
→ Piecewise[{{(Binomial[n, x]*Pochhammer[α, x]*Pochhammer[β, -x + n])/Pochhammer[α + β, n], 0 <= x <= n}}, 0]Variance[BetaBinomialDistribution[α, β, n]]
→ (n*α*β*(n + α + β))/((α + β)^2*(1 + α + β))