Beta-Binomial Distribution#

Univariate, Discrete, Bounded

The Beta-Binomial distribution is a discrete probability distribution derived from the Binomial distribution, with its probability of success in each trial governed by a Beta distribution. This distribution is characterized by three parameters: \(\alpha\), \(\beta\), and \(n\), where \(\alpha\) and \(\beta\) are the shape parameters of the Beta distribution and \(n\) is the number of trials in the Binomial distribution.

Key properties and parameters#

Support

\(x \in \{0, 1, \ldots, n\}\)

Mean

\(n \dfrac{\alpha}{\alpha + \beta}\)

Variance

\(\dfrac{n \alpha \beta (\alpha+\beta+n)}{(\alpha+\beta)^2 (\alpha+\beta+1)}\)

Parameters:

  • \(\alpha\) : (float) Shape parameter of the Beta distribution, \(\alpha > 0\).

  • \(\beta\) : (float) Shape parameter of the Beta distribution, \(\beta > 0\).

  • \(n\) : (int) Number of trials in the Binomial distribution, \(n \geq 0\).

Probability Mass Function (PMF)#

\[ f(x \mid \alpha, \beta, n) = \binom{n}{x} \frac{B(x + \alpha, n - x + \beta)}{B(\alpha, \beta)} \]

where \(B\) is the Beta function and \(\binom{n}{x}\) is the binomial coefficient.

Beta-Binomial Distribution PMF

Cumulative Distribution Function (CDF)#

\[\begin{split} F(x \mid \alpha, \beta, n) = \begin{cases} 0, & x < 0 \\ \sum_{k=0}^{x} \binom{n}{k} \frac{B(k + \alpha, n - k + \beta)}{B(\alpha, \beta)}, & 0 \leq x < n \\ 1, & x \geq n \end{cases} \end{split}\]
Beta-Binomial Distribution CDF

See also

Related Distributions:

  • Binomial - The Beta-Binomial distribution is a compound distribution derived from the Binomial distribution.

  • Beta - The Beta-Binomial distribution is a compound distribution parameterized by the Beta distribution.

References#