In a sequence of consecutive years 1, 2, . . . , T, an annual number of high-risk events is

recorded by a bank. The random number Nt of high-risk events in a given year is modelled

via Poisson(lambda) distribution. This gives a sequence of independent counts n1, n2, . . . , nT . The

prior on is Gamma(a, b) with known a > 0, b > 0

a) Determine the Bayesian estimator of the intensity with respect to quadratic loss.

b) The bank claims that the yearly intensity is less than 4. Using Bayesian hypothesis

testing with a zero-one loss, would you accept the bank's claim? Using the same data,

would you accept the claim that the yearly intensity is less than 5?

Hint: You use the R function pgamma to answer this question.