Tag Archives: Uniform Distribution

Exam C Practice Problem 15 – Examples of Frequency-Severity Models

Problem 15-A

You are given the following:

  • The annual number of claims generated for a single risk follows a Poisson distribution with mean \theta.
  • The claim size follows a uniform distribution on (0,10 \theta).
  • The risk parameter \theta is a random variable with density function \displaystyle \pi(\theta)=\frac{1}{2} \ \theta where 0<\theta<2.
  • Given \theta, the number of claims and the claim sizes are independent.

According to the records of the insurer, there are 4 claims in the amounts 5, 5, 15 and 20 within the last 5 years.

Determine the Buhlmann credibility estimate for total claim costs in the next year.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.39

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.61

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.69

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10.25

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10.69

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Problem 15-B

You are given the following:

  • The annual number of claims generated for a single risk follows a Poisson distribution with mean \theta.
  • The claim size follows a uniform distribution on (0,100 \theta).
  • The prior distribution of \theta has a uniform distribution on (0,5).
  • Given \theta, the number of claims and the claim sizes are independent.

According to the records of the insurer, there are 5 claims in the total amount of 360 within the last 3 years.

Determine the Buhlmann credibility estimate for total claim costs in the next year.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 97.50

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 164.33

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 179.33

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 212.50

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 257.33

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

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Exam C Practice Problem 14 – Examples of Limited Fluctuation Credibility

Problem 14-A

You are given the following:

  • The annual number of claims generated from a portfolio of insurance policies follows a Poisson distribution.
  • The claim size follows a uniform distribution on (0,t) where t is unknown.
  • The number of claims and the claim sizes are independent.

Using limited fluctuation credibility, how many expected claims are required to be 95% certain that actual claim costs will be within 5% of the expected claim costs?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1443

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1579

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1936

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1945

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2050

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Problem 14-B

You are given the following:

  • The annual number of claims generated from a portfolio of insurance policies follows a Poisson distribution.
  • The claim size follows a distribution with the following moment generating function.
    • \displaystyle M(t)=\frac{1}{(1-10t)^4}
  • The number of claims and the claim sizes are independent.

What is the least number of expected claims that are required to be 90% certain that actual claim costs will be within 5% of the expected claim costs?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 820

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1230

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1353

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1376

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1396

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

Exam C Practice Problem 4 – Buhlmann Credibility Examples

Problem 4-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 a uniform distribution on (0.5,2.5).

After observing for three calendar years, this risk is found to have incurred 1 claim in year 1, 2 claims in year 2 and 3 claims in year 3.

Determine the Buhlmann credibility estimate for the expected claim frequency for the given risk in year 4.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.50

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.65

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.70

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.97

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

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Problem 4-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{1}{2} \ (2-\theta), \ \ \ \ \ \ 0<\theta<2

The given risk is observed for 6 calendar years and is found to have incurred a total of 10 claims.

Determine the Buhlmann credibility estimate for the expected claim frequency for the given risk for the next calendar year.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{2}{3}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{62}{57}

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

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

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

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