An angler starts fishing at time 0. Fish bite in a Poisson Process of rate $\Lambda$ per hour, so that, if $\Lambda=\lambda$, the number $N_{t}$ of fish he catches in the first $t$ hours has the Poisson distribution $\mathcal{P}(\lambda t)$, while $T_{n}$, the time in hours until his $n$th bite, has the Gamma distribution $\Gamma(n, \lambda)$, with density function

$$p(t \mid \lambda)=\frac{\lambda^{n}}{(n-1) !} t^{n-1} e^{-\lambda t} \quad(t>0) .$$

Bystander $B_{1}$ plans to watch for 3 hours, and to record the number $N_{3}$ of fish caught. Bystander $B_{2}$ plans to observe until the 10 th bite, and to record $T_{10}$, the number of hours until this occurs.

For $B_{1}$, show that $\widetilde{\Lambda}_{1}:=N_{3} / 3$ is an unbiased estimator of $\Lambda$ whose variance function achieves the Cramér-Rao lower bound

Find an unbiased estimator of $\Lambda$ for $B_{2}$, of the form $\widetilde{\Lambda}_{2}=k / T_{10}$. Does it achieve the Cramér-Rao lower bound? Is it minimum-variance-unbiased? Justify your answers.

In fact, the 10 th fish bites after exactly 3 hours. For each of $B_{1}$ and $B_{2}$, write down the likelihood function for $\Lambda$ based their observations. What does the Likelihood Principle have to say about the inferences to be drawn by $B_{1}$ and $B_{2}$, and why? Compute the estimates $\tilde{\lambda}_{1}$ and $\widetilde{\lambda}_{2}$ produced by applying $\widetilde{\Lambda}_{1}$ and $\widetilde{\Lambda}_{2}$ to the observed data. Does the method of minimum-variance-unbiased estimation respect the Likelihood Principle?