.. _discrete-betabinom:

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 :math:0 \leq k \leq n, is:

.. math::

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 :math:B(a, b) is the Beta function.

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

.. math::

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

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

.. math::

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

Implementation: scipy.stats.betabinom