Exam C Practice Problem 5 – Bayesian Estimate of Claim Frequency

Problem 5-A

You are given the following:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The prior distribution of \theta has the following density function.
      • \displaystyle \pi(\theta)=\frac{2}{\theta^3}, \ \ \ \ \ \ 1<\theta<\infty

After observing for one calendar year, this risk is found to have incurred 4 claims.

Determine the Bayesian expected claim frequency for the given risk in the next year.

\text{ }

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{5}{2}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{7}{2}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4

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Problem 5-B

You are given the following:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The prior distribution of \theta has the following density function.
      • \displaystyle \pi(\theta)=\frac{2}{\theta^3}, \ \ \ \ \ \ 1<\theta<\infty

After observing for two calendar years, this risk is found to have incurred 2 claims in each year.

Determine the Bayesian expected claim frequency for the given risk in the next year.

\text{ }

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{4}{3}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{8}{5}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{5}{3}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{9}{5}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2

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

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