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The posterior predictive distribution for a single additional observation is a [[negative binomial distribution]] distribution,{{Citation needed|date=April 2012}} sometimes called a Gamma-Poisson distribution.
==Bivariate Poisson distribution==
This distribution has been extended to the bivariate case. The [[generating function]] for this distribution is
: <math> g( u, v ) = exp[ ( \theta_1 - \theta_12 )( u - 1 ) + ( \theta_2 - \theta_12 )( v - 1 ) + \theta_12 ( uv -1 ) ] </math>
with
: <math> \theta_1, \theta_2 > \theta_12 > 0 </math>
The marginal distributions are Poisson( ''θ''<sub>1</sub> ) and Poisson( ''θ''<sub>2</sub> ) and the correlation coefficient is limited to the range
: <math> 0 \le \rho \le min[ \frac{ \theta_1 }{ \theta_2 }, \frac{ \theta_2 }{ \theta_1 } ]</math>
== See also ==
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