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 $X$. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 400$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 405$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 450$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 475$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4045$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.16$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.18$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.21$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.23$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.29$

___________________________________________________________________________________ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

___________________________________________________________________________________

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