# Exam C Practice Problem 8 – Bayesian Estimates of Claim Frequency

Problem 8-A

A portfolio consists of ten independent risks divided into two classes. Class 1 contains 6 risks and Class 2 contains 4 risks.

The risks in each class are assumed to follow identical annual claim frequency distribution. The following table shows the distributions of the number of claims in a calendar year.

$\displaystyle \begin{bmatrix} \text{ }&\text{ }&\text{Class 1} &\text{ }&\text{Class 2} \\X=x&\text{ }&P(X=x) &\text{ }&P(X=x) \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 0&\text{ }&0.50 &\text{ }&0.20 \\ 1&\text{ }&0.25&\text{ }&0.25 \\ 2&\text{ }&0.12&\text{ }&0.30 \\ 3&\text{ }&0.08&\text{ }&0.15 \\ 4&\text{ }&0.05&\text{ }&0.10 \end{bmatrix}$

The risks in the portfolio are observed for one calendar year. The following table shows the observed results.

$\displaystyle \begin{bmatrix} \text{Number of Claims}&\text{ }&\text{Number of Risks} \\\text{ }&\text{ }&\text{ } &\text{ } \\ 0&\text{ }&4 \\ 1&\text{ }&2 \\ 2&\text{ }&2 \\ 3&\text{ }&1 \\ 4&\text{ }&1 \end{bmatrix}$

Determine the Bayesian expected number of claims per risk in the next year.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.10$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.15$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.22$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.29$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.315$

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Problem 8-B

A portfolio consists of ten independent risks divided into two classes. Both classes contain the same number of risks.

The risks in each class are assumed to follow identical annual claim frequency distribution. The following table shows the distributions of the number of claims in a calendar year.

$\displaystyle \begin{bmatrix} \text{ }&\text{ }&\text{Class 1} &\text{ }&\text{Class 2} \\X=x&\text{ }&P(X=x) &\text{ }&P(X=x) \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 0&\text{ }&0.5 &\text{ }&0.1 \\ 1&\text{ }&0.3&\text{ }&0.4 \\ 2&\text{ }&0.1&\text{ }&0.3 \\ 3&\text{ }&0.05&\text{ }&0.1 \\ 4&\text{ }&0.05&\text{ }&0.1 \end{bmatrix}$

The risks in the portfolio are observed for one calendar year. The following table shows the observed results.

$\displaystyle \begin{bmatrix} \text{Number of Claims}&\text{ }&\text{Number of Risks} \\\text{ }&\text{ }&\text{ } &\text{ } \\ 0&\text{ }&2 \\ 1&\text{ }&2 \\ 2&\text{ }&3 \\ 3&\text{ }&1 \\ 4&\text{ }&2 \end{bmatrix}$

What is the probability that a randomly selected risk in this portfolio will incur exactly 2 claims in the next year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.26$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.27$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.28$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.29$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.30$

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