**Problem 9-A**

A portfolio consists of independent risks divided into two classes. Eighty percent of the risks are in Class 1 and twenty percent are in Class 2.

The following provides more information about these risks:

- For each risk in Class 1, the number of claims in a year has a Poisson distribution with mean such that follows a Gamma distribution with mean 1.6 and variance 1.28.
- For each risk in Class 2, the number of claims in a year has a Poisson distribution with mean such that follows a Gamma distribution with mean 2.5 and variance 3.125.

An actuary is hired to examine the claim experience of the risks in this portfolio. What proportion of the risks can be expected to incur exactly 1 claim in one year?

**Problem 9-B**

A portfolio consists of independent risks divided into two classes. Sixty percent of the risks are in Class 1 and fourty percent are in Class 2.

The following provides more information about the two classes of risks:

- For each risk in Class 1, the number of claims in a year has a Poisson distribution with mean such that follows a Gamma distribution with mean 2.4 and variance .
- For each risk in Class 2, the number of claims in a year has a Poisson distribution with mean such that follows a Gamma distribution with mean 3.75 and variance .

An actuary is hired to examine the claim experience of the risks in this portfolio. Of the risks that incur exactly 2 claims in a year, what proportion of the risks can be expected to come from Class 2?

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

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