Question 4 (35 marks) A waiting time random variable X follows an exponential distribution with a rate 10 parameterized

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Question 4 35 Marks A Waiting Time Random Variable X Follows An Exponential Distribution With A Rate 10 Parameterized 1 (51.25 KiB) Viewed 43 times

Question 4 (35 marks) A waiting time random variable X follows an exponential distribution with a rate 10 parameterized as probability density function, f(x) = he** for x>0. The rate 2 has a gamma prior distribution with parameters and B. A Bayesian credibility model provides that the posterior mean of a can be expressed as: ! ZĨ+(1-2) B a where Z- and m being the numbe of past waiting times observed. a+m-1 Assume that the parameters of the prior gamma distribution are a=5 and B=1. m (a) - 1 (b) (c) Determine an estimate of the posterior mean of 1" assuming m=10 by implementing 3,000 Monte Carlo repetitions. (8 marks) Determine an estimate of the posterior mean of 1" and the value of x when m=1,000, by implementing 3,000 Monte Carlo repetitions. (8 marks) Plot the histograms of the samples of the posterior mean of 1" and of x obtained in part (b). (6 marks) Compare, by visual inspection of the graphs in part (e), the distributions of the posterior mean of 2- and the distribution of when m=1,000. (2 marks) Comment on the behavior of the credibility model as m increases with the findings in part (d). (4 marks) (d) (e)