1. A laboratory is estimating the rate of tumorigenesis (the formation of tumors) in tm strains of mice, A and B. They have tumor count data for ID 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. Tumor count rates for type B mice are unknown, but type B mice are related to type A mice. Assuming a Poisson sampling distribution for each group with rates HA and I93. Based on previous research you settle on the following prior distribution: HA ~ gamma{12, 11]}, 93 ~ gamma{12, 1) (a) Before seeing any data, which group do you expect to have a higher average incidence of cancer? Vihich group are you more certain about a priori? You answers should be based on the priors specied above. (b) After you the complete of the experiment, you observe the following tumor counts for the two populations: yA = (12,9,12, 14, 13,13,151, 3, 15, 6} yB = {11, 1]., 10, 9, 9581?: 10: 6: 8': 81 91 7) Write down the posterior distributions, posterior means, posterior variances and 95% quantilebased credible intervals for 9,1 and 6'3 (c) lCompute and plot the posterior expectation of EB given *9}; under the prior distribution gamma(12)