Exam C Practice Problem 1 – Working with Mixture Distributions

Problem 1-A

You are given:

    • The claim size X for a policyholder randomly chosen from a large group of insureds is a mixture of a Burr distribution with \alpha=1, \theta=\sqrt{8000} and \gamma=2 and a Pareto distribution with \alpha=1 and \theta=8000.
    • The mixture distribution of X has equal mixing weights.

Calculate the median of X.

\text{ }

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 400

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 405

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 450

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 475

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4045

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Problem 1-B

You are given:

    • The claim size X in the current year for a policyholder randomly chosen from a large group of insureds is a mixture of a Burr distribution with \alpha=2, \theta=\sqrt{1000} and \gamma=2 and a Pareto distribution with \alpha=2 and \theta=1000.
    • The mixture distribution of X has mixing weights 90% (for the Burr distribution) and 10% (for the Pareto distribution).
    • Suppose that the claim size for the chosen policyholder in the next year will increase 20% due to inflation.

What is the probability that the claim size in the next year will exceed 50?

\text{ }

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.16

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.18

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.21

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.23

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.29

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

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