 Solution

Use $A=a+{\sum }_i{x}_i$ and $B=b+n$ as above. The future observation is independent of the old observations conditional on $\theta$, so the posterior predictive probability is obtained by integrating its Poisson distribution against the posterior distribution of $\theta$:The last integral is the normalizing gamma function integral, equal to $\Gamma (A+k)/(B+1{)}^{A+k}$. This is a negative binomial distribution in the count-of-failures convention with shape $A$ and success probability $B/(B+1)$; the gamma-function expression remains valid for noninteger $A>0$. As checks, its mean is $A/B$, and the law of total variance gives variance $A/B+A/B^2$. The additional term is the contribution of posterior uncertainty in $\theta$.