 Solution

Write $m=r+s$ and $\beta =\lambda +n$. Conditional independence of the next count given $\theta$ allows us to average its Poisson distribution over the posterior distribution:This Poisson-gamma mixture is the posterior predictive distribution. In the stated negative binomial distribution convention its parameters areIts expected value $m/\beta$ equals the posterior mean of $\theta$, while its variance $m/\beta +m/{\beta }^2$ includes both conditional Poisson variation and uncertainty in $\theta$.