Probability mass function  
Cumulative distribution function  
Notation  

Parameters 
n ∈ N_{0} — number of trials (real) (real)  
Support  x ∈ { 0, …, n }  
PMF 
where is the beta function  
CDF 
where _{3}F_{2}(a;b;x) is the generalized hypergeometric function  
Mean  
Variance  
Skewness  
Ex. kurtosis  See text  
MGF  where is the hypergeometric function  
CF 
 
PGF 

In probability theory and statistics, the betabinomial distribution is a family of discrete probability distributions on a finite support of nonnegative integers arising when the probability of success in each of a fixed or known number of Bernoulli trials is either unknown or random. The betabinomial distribution is the binomial distribution in which the probability of success at each of n trials is not fixed but randomly drawn from a beta distribution. It is frequently used in Bayesian statistics, empirical Bayes methods and classical statistics to capture overdispersion in binomial type distributed data.
The betabinomial is a onedimensional version of the Dirichletmultinomial distribution as the binomial and beta distributions are univariate versions of the multinomial and Dirichlet distributions respectively. The special case where α and β are integers is also known as the negative hypergeometric distribution.
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 parameter in the binomial distribution as being randomly drawn from a beta distribution. Suppose we were interested in predicting the number of heads, in future trials. This is given by
Using the properties of the beta function, this can alternatively be written
The betabinomial distribution can also be motivated via an urn model for positive integer values of α and β, known as the Pólya urn model. Specifically, imagine an urn containing α red balls and β black balls, where random draws are made. If a red ball is observed, then two red balls are returned to the urn. Likewise, if a black ball is drawn, then two black balls are returned to the urn. If this is repeated n times, then the probability of observing x red balls follows a betabinomial distribution with parameters n, α and β.
If the random draws are with simple replacement (no balls over and above the observed ball are added to the urn), then the distribution follows a binomial distribution and if the random draws are made without replacement, the distribution follows a hypergeometric distribution.
The first three raw moments are
and the kurtosis is
Letting we note, suggestively, that the mean can be written as
and the variance as
where . The parameter is known as the "intra class" or "intra cluster" correlation. It is this positive correlation which gives rise to overdispersion. Note that when , no information is available to distinguish between the beta and binomial variation, and the two models have equal variances.
The factorial moments of a Betabinomial random variable X are
The method of moments estimates can be gained by noting the first and second moments of the betabinomial and setting those equal to the sample moments and . We find
These estimates can be nonsensically negative which is evidence that the data is either undispersed or underdispersed relative to the binomial distribution. In this case, the binomial distribution and the hypergeometric distribution are alternative candidates respectively.
While closedform maximum likelihood estimates are impractical, given that the pdf consists of common functions (gamma function and/or Beta functions), they can be easily found via direct numerical optimization. Maximum likelihood estimates from empirical data can be computed using general methods for fitting multinomial Pólya distributions, methods for which are described in (Minka 2003). The R package VGAM through the function vglm, via maximum likelihood, facilitates the fitting of glm type models with responses distributed according to the betabinomial distribution. There is no requirement that n is fixed throughout the observations.
The following data gives the number of male children among the first 12 children of family size 13 in 6115 families taken from hospital records in 19th century Saxony (Sokal and Rohlf, p. 59 from Lindsey). The 13th child is ignored to assuage the effect of families nonrandomly stopping when a desired gender is reached.
Males  0  1  2  3  4  5  6  7  8  9  10  11  12 
Families  3  24  104  286  670  1033  1343  1112  829  478  181  45  7 
The first two sample moments are
and therefore the method of moments estimates are
The maximum likelihood estimates can be found numerically
and the maximized loglikelihood is
from which we find the AIC
The AIC for the competing binomial model is AIC = 25070.34 and thus we see that the betabinomial model provides a superior fit to the data i.e. there is evidence for overdispersion. Trivers and Willard postulate a theoretical justification for heterogeneity in genderproneness among mammalian offspring.
The superior fit is evident especially among the tails
Males  0  1  2  3  4  5  6  7  8  9  10  11  12 
Observed Families  3  24  104  286  670  1033  1343  1112  829  478  181  45  7 
Fitted Expected (BetaBinomial)  2.3  22.6  104.8  310.9  655.7  1036.2  1257.9  1182.1  853.6  461.9  177.9  43.8  5.2 
Fitted Expected (Binomial p = 0.519215)  0.9  12.1  71.8  258.5  628.1  1085.2  1367.3  1265.6  854.2  410.0  132.8  26.1  2.3 
The betabinomial distribution plays a prominent role in the Bayesian estimation of a Bernoulli success probability . Let be a sample of independent and identically distributed Bernoulli random variables . Suppose, our knowledge of ,  in Bayesian fashion  is uncertain and is modeled by the prior distribution . If then through compounding, the prior predictive distribution of
After observing we note that the posterior distribution for
where is a normalizing constant. We recognize the posterior distribution as a .
Thus, again through compounding, we find that the posterior predictive distribution of a sum of a future sample of size of random variables is
To draw a betabinomial random variate simply draw a and then draw .