**Problem 25-A**

You are given the following information about an insured population:

- For each risk in the population, the annual number of claims follows a Poisson distribution with mean .
- The parameter follows a Gamma distribution with mean 2 and variance 0.8.

A risk is randomly selected from the insured population. After observing the selected risk for 10 years, the following is known.

- Based on the number of claims observed in the first 9 years, the Bayesian estimate for the expected number of claims per year for this risk is 4.
- Based on all the claims observed in the entire 10-year period, the Bayesian estimate for the expected number of claims per year for this risk is 3.92.

What is the number of claims observed in the last year of the observation period for the selected risk?

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**Problem 25-B**

You are given the following information about an insured population:

- For each risk in the population, the annual number of claims follows a Poisson distribution with mean .
- The parameter follows a Gamma distribution with mean 1.6 and variance 0.32.

A risk is randomly selected from the insured population. After observing the selected risk for 10 years, the following is known.

- Based on the number of claims observed in the first 5 years, the Bayesian estimate for the expected number of claims per year for this risk is 1.7.
- Based on all the claims observed in the entire 10-year period, the Bayesian estimate for the expected number of claims per year for this risk is 1.8.

What is the number of claims observed in the last 5 years of the observation period for the selected risk?

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Tagged: Actuarial Exam, Bayesian Credibility, Bayesian Probability, CAS Exam 4, CAS Exam 4 Practice Problem, Frequency, Gamma Distribution, Loss Models, Poisson Claim Frequency, Poisson Distribution, Poisson-Gamma Model, Probability, SOA Exam C, SOA Exam C Practice Problem, Variance

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