Exam P Practice Problem 48 – Dental and Vision Expenses

Problem 48-A

An insurance company sells an ancillary health benefit plan that reimburses dental expenses and vision care expenses to the plan members. The plan provides no other benefits in addition to dental and vision care.

For the basic plan, the annual amount of dental reimbursement and the annual amount of vision care reimbursement are identically and exponentially distributed with mean 2 (in hundreds).

An actuary is designing a deluxe ancillary plan that provides similar but richer benefits. The annual amount of deluxe dental reimbursement is four times that of the basic plan. The annual amount of deluxe vision care reimbursement is two times that of the basic plan. Except for the richer benefit amounts, the actuary believes that the deluxe plan reimbursements have the same underlying probability distribution as the basic plan.

For both the basic plan and deluxe plan, the annual amount of dental reimbursement is independent of the annual amount of vision care reimbursement.

Which of the following is the probability density function of the total annual amount of expenses reimbursed by the deluxe plan?

$\displaystyle (A) \ \ \ \ \ \ f(x)=\frac{1}{64 \times 5!} \ x^5 \ e^{ -0.5 \ x}$

$\displaystyle (B) \ \ \ \ \ \ f(x)=4 \ e^{-0.125 \ x} - 4 \ e^{-0.25 \ x}$

$\displaystyle (C) \ \ \ \ \ \ f(x)=0.125 \ e^{-0.125 \ x} + 0.25 \ e^{-0.25 \ x}$

$\displaystyle (D) \ \ \ \ \ \ f(x)=0.25 \ e^{-0.125 \ x} - 0.25 \ e^{-0.25 \ x}$

$\displaystyle (E) \ \ \ \ \ \ f(x)=1.25 \ e^{-0.25 \ x} - 0.5 \ e^{-0.125 \ x}$

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Problem 48-B

When tornadoes occur, the total annual amount of property damages due to tornadoes (in millions) in area A has an exponential distribution with mean 20.

When tornadoes occur, the total annual amount of property damages due to tornadoes (in millions) in area B has an exponential distribution with mean 25.

Since area A and area B are sufficiently far apart, assume that the amount of tornado damages in one area is independent of the amount of damages in the other area.

What is the probability density function of the total annual amount of tornado damages for these two areas?

$\displaystyle (A) \ \ \ \ \ \ f(x)= e^{-0.04 \ x} - \ e^{-0.05 \ x}$

$\displaystyle (B) \ \ \ \ \ \ f(x)=0.2 \ e^{-0.04 \ x} - 0.2 \ e^{-0.05 \ x}$

$\displaystyle (C) \ \ \ \ \ \ f(x)=0.04 \ e^{-0.04 \ x} + 0.05 \ e^{-0.05 \ x}$

$\displaystyle (D) \ \ \ \ \ \ f(x)=\frac{1}{45} \ e^{- \frac{1}{45} \ x}$

$\displaystyle (E) \ \ \ \ \ \ f(x)=0.3 \ e^{-0.05 \ x} - 0.2 \ e^{-0.04 \ x}$

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