Exam C Practice Problem 17 – Estimating Claim Frequency

Both Problems 17-A and 17-B use the following information.

An insurance portfolio consists of independent risks.

For each risk in this portfolio, the number of claims in a year has a Poisson distribution with mean \theta. The parameter \theta follows a Gamma distribution.

A risk is randomly selected from this portfolio. Prior to obtaining any claim experience, the number of claims in a year for this risk has a distribution with mean 0.6 and variance 0.72.

After observing this risk for one year, insurance company records indicate that there are 2 claims for this risk.

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Problem 17-A

After knowing the insurance company records, what is the expected number of claims per year for this risk?

\text{ }

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.60

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.83

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.91

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.25

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.00

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Problem 17-B

After knowing the insurance company records, what is the variance of the number of claims per year for this risk?

\text{ }

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{26}{36}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{32}{36}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{35}{36}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{42}{36}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{45}{36}

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

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