## Exam C Practice Problem 25 – Estimation Based on Claim Experience

Problem 25-A

You are given the following information about an insured population:

• For each risk in the population, the annual number of claims follows a Poisson distribution with mean $\theta$.
• The parameter $\theta$ follows a Gamma distribution with mean 2 and variance 0.8.

A risk is randomly selected from the insured population. After observing the selected risk for 10 years, the following is known.

• Based on the number of claims observed in the first 9 years, the Bayesian estimate for the expected number of claims per year for this risk is 4.
• Based on all the claims observed in the entire 10-year period, the Bayesian estimate for the expected number of claims per year for this risk is 3.92.

What is the number of claims observed in the last year of the observation period for the selected risk?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7$

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Problem 25-B

You are given the following information about an insured population:

• For each risk in the population, the annual number of claims follows a Poisson distribution with mean $\theta$.
• The parameter $\theta$ follows a Gamma distribution with mean 1.6 and variance 0.32.

A risk is randomly selected from the insured population. After observing the selected risk for 10 years, the following is known.

• Based on the number of claims observed in the first 5 years, the Bayesian estimate for the expected number of claims per year for this risk is 1.7.
• Based on all the claims observed in the entire 10-year period, the Bayesian estimate for the expected number of claims per year for this risk is 1.8.

What is the number of claims observed in the last 5 years of the observation period for the selected risk?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 12$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 16$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 19$

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

## Exam C Practice Problem 24 – Bayesian Credibility Example

Both Problems 24-A and 24-B use the following information.

You are given the following information:

• The claim size of an insured has an exponential distribution with mean $\displaystyle \frac{1}{\theta}$.
• The parameter $\theta$ has a Gamma distribution with mean 6 and variance 12.

A randomly selected insured has one claim of size 10.

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Problem 24-A

What is the Bayesian estimate of the expected amount of the next claim for this insured?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.5$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.2$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.1$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.5$

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Problem 24-B

What is the posterior probability that the size of the next claim for this insured will be greater than 5?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.15$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.19$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.21$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25$

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

## Exam C Practice Problem 23 – Working with Credibility Estimates

Both Problems 23-A and 23-B use the following information.

You are given the following:

• A portfolio of independent risks is divided into two classes.
• Each class contains the same number of risks.
• For each risk in Class 1, the claim size follows a zero-truncated geometric distribution with mean 1.5.
• For each risk in Class 2, the claim size follows a zero-truncated geometric distribution with mean 2.5.
• See definition of zero-truncated distribution here.

A risk is selected at random from the portfolio. The first claim observed for this risk is 3.

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Problem 23-A

Calculate the Bayesian credibility estimate of the expected value of the next claim that will be observed for this risk.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.00$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.10$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.16$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.20$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.30$

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Problem 23-B

Calculate the Buhlmann credibility estimate of the expected value of the next claim that will be observed for this risk.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.00$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.10$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.16$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.20$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.30$

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

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

## 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?

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

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

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

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

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

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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?

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

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

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

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

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

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

## Exam C Practice Problem 13 – Working with Aggregate Claims

Both Problems 13-A and 13-B use the following information.

A portfolio if independent risks is divided into two classes. Sixty five percent of the risks are in Class 1 and thirty five percent are in Class 2.

The risks in each class are assumed to follow identical annual aggregate claim distribution. The following shows the aggregate claim distributions for the two classes.

$\displaystyle \begin{bmatrix} X=x&\text{ }&P(X=x \lvert \text{Class 1}) &\text{ }&P(X=x \lvert \text{Class 2}) \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 0&\text{ }&\displaystyle \frac{24}{40} &\text{ }&\displaystyle \frac{4}{12} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 5&\text{ }&\displaystyle \frac{6}{40}&\text{ }&\displaystyle \frac{2}{12} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 10&\text{ }&\displaystyle \frac{7}{40}&\text{ }&\displaystyle \frac{3}{12} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 15&\text{ }&\displaystyle \frac{2}{40}&\text{ }&\displaystyle \frac{2}{12} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 20&\text{ }&\displaystyle \frac{1}{40}&\text{ }&\displaystyle \frac{1}{12} \end{bmatrix}$

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Problem 13-A

A risk is randomly selected from this portfolio and is observed to have 15 in aggregate claims in the first year.

What is the probability that the chosen risk will have 15 in aggregate claims in the second year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.09$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.10$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.11$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.13$

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Problem 13-B

A risk is randomly selected from this portfolio and is observed to have 15 in aggregate claims in the first year and 10 in aggregate claims in the second year.

What is the probability that the chosen risk will have 15 in aggregate claims in the third year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.09$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.10$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.11$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.13$

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

## Exam C Practice Problem 11 – Estimating Claim Frequency

Both Problems 11-A and 11-B use the following information.

A portfolio consists of independent risks divided into two classes. Seventy percent of the risks are in Class 1 and thirty 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 1.
• For each risk in Class 2, the number of claims in a year has a Poisson distribution with mean 2.5.

A randomly selected risk from this portfolio has 2 claims in year 1 and 2 claims in year 2.

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Problem 11-A

What is the Bayesian estimate of the number of claims in the next year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.65$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.66$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.67$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.68$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.75$

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Problem 11-B

What is the Buhlmann estimate of the number of claims in the next year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.65$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.66$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.67$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.68$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.75$

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

## Exam C Practice Problem 10 – Examples of Claim Frequency Models

Problem 10-A

A portfolio consists of independent risks. For each risk, the number of claims in a year has a Poisson distribution with mean $\lambda$. The parameter $\lambda$ is a mixture of a Gamma distribution with mean 1.6 and variance 1.28 (80% weight) and a Gamma distribution with mean 2.5 and variance 3.125 (20% weight).

A risk is randomly selected from this portfolio and observed for 3 years and is found to have incurred 4 claims. What is the probability that this risk will incur exactly 1 claim in the upcoming year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.267$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.285$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.303$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.319$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.357$

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Problem 10-B

A portfolio consists of independent risks. For each risk, the number of claims in a year has a Poisson distribution with mean $\lambda$. The parameter $\lambda$ is a mixture of a Gamma distribution with mean 2.4 and variance $\displaystyle \frac{48}{25}$ (60% weight) and a Gamma distribution with mean 3.75 and variance $\displaystyle \frac{75}{16}$ (40% weight).

A risk is randomly selected from this portfolio and observed for 2 years and is found to have incurred 3 claims.

If this risk incurs exactly 2 claims in the upcoming year, what is the probability that the given risk is from Class 2?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.205$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.214$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.263$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.275$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.300$

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Revised: May 1, 2016.
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$\copyright \ 2013-2016 \ \ \text{Dan Ma}$

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

## Exam c Practice Problem 7 – Working with Buhlmann Credibility

Problem 7-A

You are given the following information:

• The number of claims in a calendar year for a given risk follows a Poisson distribution with mean $\theta$.
• The risk parameter $\theta$ follows a Gamma distribution whose coefficient of variation is 0.5.
• After observing the given risk for 5 calendar years, twelve claims are observed.
• Based on the observed data, the posterior distribution of $\theta$ is a continuous distribution whose mean is 2.0.

What is the Buhlmann credibility used in estimating the expected claim frequency for the given risk in the next period?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.500$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.600$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.625$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.650$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.667$

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Problem 7-B

You are given the following information:

• The number of claims in a calendar year for a given risk follows a Poisson distribution with mean $\theta$.
• The risk parameter $\theta$ follows a Gamma distribution with mean 1.5.
• The value of Buhlmann’s k is 8.
• After observing the given risk for 4 calendar years, the posterior distribution of $\theta$ is a continuous distribution whose mean is 1.75

What is the number of claims observed in the observation period?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 11$

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