 Beta-binomial distribution

Mix a binomial distribution with a beta distribution for its success probability: $\Theta \sim \mathrm{Beta}(\alpha ,\beta )$ and $Y\mid \Theta \sim \mathrm{Binomial}(n,\Theta )$. Integrating their densities gives the displayed mass for $y=0,\dots ,n$. Its expectation is $n\alpha /(\alpha +\beta )$, and its variance is $n\alpha \beta (\alpha +\beta +n)/[(\alpha +\beta {)}^2(\alpha +\beta +1)]$. It is the posterior predictive distribution for repeated Bernoulli trials after a beta-prior update.