# Beta-Binomial Distribution

The beta-binomial distribution is a binomial distribution with a probability of success p that follows a beta distribution. The probability mass function for betabinom, defined for \(0 \leq k \leq n\), is:

\[f(k; n, a, b) = \binom{n}{k} \frac{B(k + a, n - k + b)}{B(a, b)}\]

for `k` in `{0, 1,..., n}`, where \(B(a, b)\) is the Beta function.

In the limiting case of \(a = b = 1\), the beta-binomial distribution reduces to a discrete uniform distribution:

\[f(k; n, 1, 1) = \frac{1}{n + 1}\]

In the limiting case of \(n = 1\), the beta-binomial distribution reduces to a Bernoulli distribution with the shape parameter \(p = a / (a + b)\):

\[\begin{split}f(k; 1, a, b) = \begin{cases}a / (a + b) & \text{if}\; k = 0 \\b / (a + b) & \text{if}\; k = 1\end{cases}\end{split}\]

Implementation: `scipy.stats.betabinom`