We will use the Minitab macro PoisGamP, or poisgamp function in R, to nd the posterior distribution of the Poisson probability when we have a random sample of observations from a Poisson( ) distribution and we have a \(\operatorname{gamma}(r v)\) prior for . The gamma family of priors is the conjugate family for Poisson observations. That means that if we start with one member of the family as the prior distribution, we will get another member of the family as the posterior distribution. The simple updating rules are add sum of observations to \(r\) and add sample size to \(v\). When we start with a \(\operatorname{gamma}\left(\begin{array}{r}r\end{array}\right)\) prior, we get a \(\operatorname{gamma}(r \quad v)\) posterior where \(r=r+\quad(y)\) and \(v=v+n\).
Suppose we have a random sample of ve observations from a Poisson( ) distribution. They are:

(a) Suppose we start with a positive uniform prior for . What \(\operatorname{gamma}(r v)\) prior will give this form?
(b) [Minitab:] Find the posterior distribution using the Minitab macro PoisGamP or the \(\mathrm{R}\) function poisgamp.
[R:] Find the posterior distribution using the \(\mathrm{R}\) function poisgamp.
(c) Find the posterior mean and median.
(d) Find a 95\% Bayesian credible interval for .

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