**QUESTION**

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3.3 Tumor counts: A cancer laboratory is estimating the rate of tumorigenesis in two strains of mice, $A$ and $B$. They have tumor count data for 10 mice in strain $A$ and 13 mice in strain $B$. Type $A$ mice have been well studied, and information from other laboratories suggests that type $A$ mice have tumor counts that are approximately Poisson-distributed with a mean of 12. Tumor count rates for type $B$ mice are unknown, but type $B$ mice are related to type $A$ mice. The observed tumor counts for the two populations are \[ \begin{array}{c} \boldsymbol{y}_{A}=(12,9,12,14,13,13,15,8,15,6) ; \\ \boldsymbol{y}_{B}=(11,11,10,9,9,8,7,10,6,8,8,9,7) . \end{array} \] a) Find the posterior distributions, means, variances and $95 \%$ quantilebased confidence intervals for $\theta_{A}$ and $\theta_{B}$, assuming a Poisson sampling distribution for each group and the following prior distribution: \[ \theta_{A} \sim \operatorname{gamma}(120,10), \theta_{B} \sim \operatorname{gamma}(12,1), p\left(\theta_{A}, \theta_{B}\right)=p\left(\theta_{A}\right) \times p\left(\theta_{B}\right) . \] b) Compute and plot the posterior expectation of $\theta_{B}$ under the prior distribution $\theta_{B} \sim \operatorname{gamma}\left(12 \times n_{0}, n_{0}\right)$ for each value of $n_{0} \in\{1,2, \ldots, 50\}$. Describe what sort of prior beliefs about $\theta_{B}$ would be necessary in order for the posterior expectation of $\theta_{B}$ to be close to that of $\theta_{A}$. c) Should knowledge about population $A$ tell us anything about population $B$ ? Discuss whether or not it makes sense to have $p\left(\theta_{A}, \theta_{B}\right)=$ $p\left(\theta_{A}\right) \times p\left(\theta_{B}\right)$.