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We write ? ∼ Poisson (?) if ? has the Poisson distribution with rate ? >...

We write ? ∼ Poisson (?) if ? has the Poisson distribution with rate ? > 0, that is, its p.m.f. is ?(?|?) = Poisson(?|?) = ? ^??^x /?!

Assume a gamma distribution as the prior for ? where ?(?) = ? ^??(?) ? ^?-1e ^??

?> 0 Use Bayes Rule to derive the posterior distribution ?(?|?).

b. Let’s reconsider the car accidents example introduced in classed. Suppose that (X) the number of car accidents at a fixed point on a different road within a fixed time window (say one month) is 4. Suppose also that we know this road well and our prior impression of the number of car accidents is that there is on average 1 accident per month with variance 1. The following data have collected over a sample of 5 months where the number of crashes observed were 4, 5, 7, 8, 4. In light of these data (n=5) what your estimate of ? now be?

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