In the situation of Exercise 9, suppose that a prior distribution is used for with p.d.f.

In the situation of Exercise 9, suppose that a prior distribution is used for θ with p.d.f. ξ(θ) = 0.1 exp(ˆ’0.1θ) for θ >0. (This is the exponential distribution with parameter 0.1.)
a. Prove that the posterior p.d.f. of θ given the data observed in Exercise 9 is

b. Calculate the posterior probability that |θ ˆ’ 2| 2 is the observed average of the data values.
c. Calculate the posterior probability that θ is in the confidence interval found in part (a) of Exercise 9.
d. Can you explain why the answer to part (b) is so close to the answer to part (e) of Exercise 9? Compare the posterior p.d.f. in part (a) to the function in Eq. (8.5.15).

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įf 4.8<θ<5.2. otherwise. 4.122 exp(-0.18)