 Solution

The posterior predictive probability averages the conditional probability over the Bayesian posterior. Using its gamma distribution density,For fixed $a,b$ and $n/T\to \hat{\lambda }<\infty$,Consequently $p_0/e^{-n/T}\to 1$, and both tend to $e^{-\hat{\lambda }}$. This is reasonable because the posterior mean tends to the observed rate and the posterior variance tends to zero. Averaging $e^{-\lambda }$ then approaches evaluating it at the estimated rate: with abundant observations, predictive uncertainty about the rate vanishes.