**Problem 10-A**

A portfolio consists of independent risks. For each risk, the number of claims in a year has a Poisson distribution with mean . The parameter is a mixture of a Gamma distribution with mean 1.6 and variance 1.28 (80% weight) and a Gamma distribution with mean 2.5 and variance 3.125 (20% weight).

A risk is randomly selected from this portfolio and observed for 3 years and is found to have incurred 4 claims. What is the probability that this risk will incur exactly 1 claim in the upcoming year?

**Problem 10-B**

A portfolio consists of independent risks. For each risk, the number of claims in a year has a Poisson distribution with mean . The parameter is a mixture of a Gamma distribution with mean 2.4 and variance (60% weight) and a Gamma distribution with mean 3.75 and variance (40% weight).

A risk is randomly selected from this portfolio and observed for 2 years and is found to have incurred 3 claims.

If this risk incurs exactly 2 claims in the upcoming year, what is the probability that the given risk is from Class 2?

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Revised: May 1, 2016.

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