A beta negative binomial distribution is a discrete distribution obtained by mixing a negative
binomial distribution with a beta distribution
for its success probability 
. For positive real
numbers 
, 
,
and 
, its probability
mass function is
 |
(1)
|
for nonnegative integers 
, where 
is the beta function
and 
is a binomial coefficient. For integer 
, this models the number of failures before the 
th
success when 
has the beta distribution
with parameters 
and 
.
The mean is
 |
(2)
|
for 
, and the variance
is
 |
(3)
|
for 
.
The distribution is implemented in the Wolfram Language as BetaNegativeBinomialDistribution[alpha, beta, r].
