You are given prior hyperparameters for a Poisson process, and observational data. Find the posterior distribution for the rate parameter \(\lambda\).

(a) Prior: \(\kappa_{0}=0, \tau_{0}=0\). Observed: \(n=7\) occurrences during \(t=5\) units.

(b) Prior: \(\kappa_{0}=5, \tau_{0}=10\). Observed: \(n=3\) occurrences during \(t=7\) units.

(c) Prior: \(\lambda \sim \gamma_{(12,3)}\). Observed: \(n=17\) occurrences during \(t=5\) units. \(k=2\).

(d) Prior: \(\kappa_{0}=3, \tau_{0}=73\). Observed: \(n=6\) occurrences during \(t=119\) units.