A beta binomial distribution is a discrete distribution obtained by mixing a binomial distribution
with a beta distribution for its probability
parameter 
.
For positive parameters 
and 
and a nonnegative integer 
,
its probability mass function for 
,
..., 
is
 |
(1)
|
where 
is the beta function and 
is a binomial coefficient.
For integer 
with 
, the distribution
function is
 |
(2)
|
where 
is a gamma function and
 |
(3)
|
is a terminating generalized hypergeometric function. For real arguments between 0 and 
, 
, where 
is the floor function.
The endpoint values are 
for 
and 
for 
.
It is implemented as BetaBinomialDistribution[alpha, beta, n].
The first few raw moments are
 |  |  |
(4)
|
 |  | ![(nalpha[n(1+alpha)+beta])/((alpha+beta)(1+alpha+beta))](/images/equations/BetaBinomialDistribution/Inline24.svg) |
(5)
|
 |  | ![(nalpha[n^2(1+alpha)(2+alpha)+3n(1+alpha)beta+beta(beta-alpha)])/((alpha+beta)(1+alpha+beta)(2+alpha+beta)),](/images/equations/BetaBinomialDistribution/Inline27.svg) |
(6)
|
giving the mean and variance as
 |  |  |
(7)
|
 |  |  |
(8)
|
