## Exam C Practice Problem 18 – Estimating Claim Frequency

Problem 18-A

A portfolio of independent risks is divided into five distinct classes that are equal in size.

The annual claim count distribution for any risk in this portfolio is assumed to be a binomial distribution. The following table shows more information about these five classes.

$\displaystyle \begin{bmatrix} \text{Class}&\text{ }&\text{ }&\text{Mean} &\text{ }&\text{ }&\text{Variance} \\\text{ }&\text{ }&\text{ }&\text{Of Claim Count} &\text{ }&\text{ }&\text{Of Claim Count} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 1&\text{ }&\text{ }&\displaystyle \frac{1}{2} &\text{ }&\text{ }&\displaystyle \frac{1}{4} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 2&\text{ }&\text{ }&\displaystyle 1&\text{ }&\text{ }&\displaystyle \frac{1}{2} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 3&\text{ }&\text{ }&\displaystyle \frac{3}{2}&\text{ }&\text{ }&\displaystyle \frac{3}{4} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 4&\text{ }&\text{ }&\displaystyle 2&\text{ }&\text{ }&\displaystyle 1 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 5&\text{ }&\text{ }&\displaystyle \frac{5}{2}&\text{ }&\text{ }&\displaystyle \frac{5}{4} \end{bmatrix}$

A risk is randomly selected from this portfolio and is observed to have one claim in the last year.

What is the probability that the mean number of claims in a year for this risk is greater than 1.5?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.209$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.228$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.600$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.761$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.781$

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Problem 18-B

A portfolio of independent risks is divided into five distinct classes that are equal in size.

The annual claim count distribution for any risk in this portfolio is assumed to be a geometric distribution. The following table shows more information about these five classes.

$\displaystyle \begin{bmatrix} \text{Class}&\text{ }&\text{ }&\text{Mean} &\text{ }&\text{ }&\text{Variance} \\\text{ }&\text{ }&\text{ }&\text{Of Claim Count} &\text{ }&\text{ }&\text{Of Claim Count} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 1&\text{ }&\text{ }&\displaystyle 1 &\text{ }&\text{ }&\displaystyle 2 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 2&\text{ }&\text{ }&\displaystyle 2&\text{ }&\text{ }&\displaystyle 6 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 3&\text{ }&\text{ }&\displaystyle 3&\text{ }&\text{ }&\displaystyle 12 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 4&\text{ }&\text{ }&\displaystyle 4&\text{ }&\text{ }&\displaystyle 20 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 5&\text{ }&\text{ }&\displaystyle 5&\text{ }&\text{ }&\displaystyle 30 \end{bmatrix}$

A risk is randomly selected from this portfolio and is observed to have one claim in the last year.

What is the probability that the mean number of claims in a year for this risk is greater than 2.5?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.49$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.51$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.55$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.57$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.60$

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