Exam C Practice Problem 9 – Examples of Claim Frequency Models

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 \theta such that \theta 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 \delta such that \delta 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?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.24

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.26

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.27

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.28

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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 \theta such that \theta follows a Gamma distribution with mean 2.4 and variance \displaystyle \frac{48}{25}.
  • For each risk in Class 2, the number of claims in a year has a Poisson distribution with mean \delta such that \delta follows a Gamma distribution with mean 3.75 and variance \displaystyle \frac{75}{16}.

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?

\text{ }

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.35

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.36

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.37

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.38

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\copyright \ 2013 \ \ \text{Dan Ma}

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