You are given posterior distributions or hyperparameters for the rate \(\lambda\) of a Poisson process, and values \(k, l, m\). Calculate the predictive distribution for \(N_{+}\)and \(T_{+k}\), and calculate \(P\left(T_{+k} \leq l\right)\) and \(P\left(N_{+1} \leq m\right)\).

(a) Posterior hyperparameters: \(\kappa_{1}=7, \tau_{1}=5 . k=2, l=3, m=4\).

(b) Posterior: \(\lambda \sim \gamma_{(8,17)} \cdot k=1, l=2, m=1\).

(c) Posterior hyperparameters: \(\kappa_{1}=29, \tau_{1}=8 . k=3, l=1, m=5\).

(d) Posterior: \(\lambda \sim \gamma_{(9,192)} \cdot k=2, l=30, m=0\).