# Exam C Practice Problem 10 – Examples of Claim Frequency Models

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 $\lambda$. The parameter $\lambda$ 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? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.267$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.285$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.303$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.319$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.357$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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 $\lambda$. The parameter $\lambda$ is a mixture of a Gamma distribution with mean 2.4 and variance $\displaystyle \frac{48}{25}$ (60% weight) and a Gamma distribution with mean 3.75 and variance $\displaystyle \frac{75}{16}$ (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? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.205$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.214$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.263$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.275$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.300$

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Revised: May 1, 2016.
___________________________________________________________________________________ $\copyright \ 2013-2016 \ \ \text{Dan Ma}$