# Beta-Binomial Probability Distribution Family

A Beta-Binomial Probability Distribution Family is a binomial distribution family (of beta-binomial functions) in which the probability of success at each trial is not fixed but random and follows the beta distribution.

**See:**Discrete Probability Distribution Function, Beta-Binomial Function, Binomial Distribution, Dirichlet Distribution, Generalized Hypergeometric Series, Bernoulli Trial, Beta Distribution, Empirical Bayes Methods.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Beta-binomial_distribution Retrieved:2015-6-21.
- In probability theory and statistics, the
**beta-binomial distribution**is a family of discrete probability distributions on a finite support of non-negative integers arising when the probability of success in each of a fixed or known number of Bernoulli trials is either unknown or random. The beta-binomial distribution is the binomial distribution in which the probability of success at each trial is not fixed but random and follows the beta distribution. It is frequently used in Bayesian statistics, empirical Bayes methods and classical statistics as an overdispersed binomial distribution.It reduces to the Bernoulli distribution as a special case when

*n*= 1. For*α*=*β*= 1, it is the discrete uniform distribution from 0 to*n*. It also approximates the binomial distribution arbitrarily well for large*α*and*β*. The beta-binomial is a one-dimensional version of the Dirichlet-multinomial distribution, as the binomial and beta distributions are special cases of the multinomial and Dirichlet distributions, respectively.

- In probability theory and statistics, the

### 2011

- http://en.wikipedia.org/wiki/Beta-binomial_distribution#Maximum_likelihood_estimation
- In probability theory and statistics, the '
*beta-binomial distribution is a family of discrete probability distributions on a finite support arising when the probability of success in each of a fixed or known number of Bernoulli trials is either unknown or random. It is frequently used in Bayesian statistics, empirical Bayes methods and classical statistics as an overdispersed binomial distribution.**It reduces to the Bernoulli distribution as a special case when*n = 1. For*α*=*β*= 1, it is the discrete uniform distribution from 0 to*n*. It also approximates the binomial distribution arbitrarily well for large*α*and*β*. The beta-binomial is a one-dimensional version of the multivariate Pólya distribution, as the binomial and beta distributions are special cases of the multinomial and Dirichlet distributions, respectively.The Beta distribution is a conjugate distribution of the binomial distribution. This fact leads to an analytically tractable compound distribution where one can think of the [math]\displaystyle{ p }[/math] parameter in the binomial distribution as being randomly drawn from a beta distribution. Namely, if :[math]\displaystyle{ \begin{align} L(k|p) & = \operatorname{Bin}(k,p) \\ & = {n\choose k}p^k(1-p)^{n-k} \end{align} }[/math] is the binomial distribution where

*p*is a random variable with a beta distribution :[math]\displaystyle{ \begin{align} \pi(p|\alpha,\beta) & = \mathrm{Beta}(\alpha,\beta) \\ & = \frac{p^{\alpha-1} (1-p)^{\beta-1}} {\mathrm{B}(\alpha,\beta)} \end{align} }[/math]

- In probability theory and statistics, the '