 Solution

The MLE is $\hat{\theta }=n/S$, $S=\sum X_i$. Since $S\sim \Gamma (n,\theta )$, $\theta S\sim \Gamma (n,1)$ (in general $\gamma Y\sim \Gamma (\beta ,\lambda /\gamma )$). If $q_r$ is the $r$-quantile of $\Gamma (n,1)$, a central confidence interval is $[(q_{\alpha /2}/n)\hat{\theta },(q_{1-\alpha /2}/n)\hat{\theta }]$. A $\Gamma (\beta ,\lambda )$ prior yields posterior $\Gamma (\beta +n,\lambda +S)$ and mean $\tilde{\theta }=(\beta +n)/(\lambda +S)$. With $q_r\prime$ the quantiles of $\Gamma (\beta +n,1)$, take $l\prime =q_{\alpha /2}\prime /(\beta +n)$ and $u\prime =q_{1-\alpha /2}\prime /(\beta +n)$.