Beta negative binomial distribution

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, a beta negative binomial distribution is the <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> of a <a href="/facts/Discrete_probability_distribution/EpsKKVRu">discrete</a> <a href="/facts/Random_variable/TwTBXnLT">random variable</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 equal to the number of failures needed to get 
 
 
 
 r
 
 
 {\displaystyle r}
 
 successes in a sequence of <a href="/facts/Independence_(probability_theory)/NUzQtnUL">independent</a> <a href="/facts/Bernoulli_trial/7fm8snCf">Bernoulli trials</a>. The probability 
 
 
 
 p
 
 
 {\displaystyle p}
 
 of success on each trial stays constant within any given experiment but varies across different experiments following a <a href="/facts/Beta_distribution/DbJk4eeV">beta distribution</a>. Thus the distribution is a <a href="/facts/Compound_probability_distribution/ccJIMBxT">compound probability distribution</a>.
This distribution has also been called both the inverse Markov-Pólya distribution and the generalized Waring distribution or simply abbreviated as the BNB distribution. A shifted form of the distribution has been called the beta-Pascal distribution.
If parameters of the beta distribution are 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 and 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
, and if

X
        ∣
        p
        ∼
        
          N
          B
        
        (
        r
        ,
        p
        )
        ,
      
    
    {\displaystyle X\mid p\sim \mathrm {NB} (r,p),}

where

p
        ∼
        
          
            B
          
        
        (
        α
        ,
        β
        )
        ,
      
    
    {\displaystyle p\sim {\textrm {B}}(\alpha ,\beta ),}

then the marginal distribution of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (i.e. the <a href="/facts/Posterior_predictive_distribution/I6hSVx66">posterior predictive distribution</a>) is a beta negative binomial distribution:

X
        ∼
        
          B
          N
          B
        
        (
        r
        ,
        α
        ,
        β
        )
        .
      
    
    {\displaystyle X\sim \mathrm {BNB} (r,\alpha ,\beta ).}

In the above, 
 
 
 
 
 N
 B
 
 (
 r
 ,
 p
 )
 
 
 {\displaystyle \mathrm {NB} (r,p)}
 
 is the <a href="/facts/Negative_binomial_distribution/znzj365Y">negative binomial distribution</a> and 
 
 
 
 
 
 B
 
 
 (
 α
 ,
 β
 )
 
 
 {\displaystyle {\textrm {B}}(\alpha ,\beta )}
 
 is the <a href="/facts/Beta_distribution/DbJk4eeV">beta distribution</a>.

Beta negative binomial distribution open-in-new

Beta negative binomial distribution