This question continues exercise sheet 5, question 4. Now there are two machines, each with a different precision of measurement, 71 and 72. They each take a number of measurements with the same known mean μ = 1000. The measurements are x = (₁,...,m) on the first machine can be modelled as a random sample from normal distribution with known mean = 1000 and precision 71 and y = (y₁, yn) on the second machine can be modelled as a random sample from a normal distribution with known mean μ = 1000 and precision 72. The observed data are m= 10, (₁-1000)²=0.12, n = 8 and 1(-1000)² = 0.09. Use independent gamma prior distributions for and 2 with same parameters a and 8 as for r in exercise sheet 5, that is a = 5 and 3 = 0.05. (a) Find the joint posterior density of 71 and 72. (b) What are the marginal posterior distributions for 7 and 7₂?

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