## 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 $X$.

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

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

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

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

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

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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?

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

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

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

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

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

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