The cumulative distribution function for health care costs experienced by a policyholder is modeled by the function

[[math]] F(x) = \begin{cases} 1 - e^{-\frac{x}{100}}, \, x \gt 0 \\ 0, \, \textrm{Otherwise.} \end{cases} [[/math]]

The policy has a deductible of 20. An insurer reimburses the policyholder for 100% of health care costs between 20 and 120. Health care costs above 120 are reimbursed at 50%. Let [math]G[/math] be the cumulative distribution function of reimbursements given that the reimbursement is positive.

Calculate [math]G(115)[/math].

• 0.683
• 0.727
• 0.741
• 0.757
• 0.777

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: B

Let Y be the reimbursement. Then, G(115) = P[Y < 115 | X > 20]. For Y to be 115, the costs must be above 120 (up to 120 accounts for a reimbursement of 100). The extra 15 requires 30 in additional costs. Therefore, we need

[[math]] \begin{align*} \operatorname{P}[X \leq 150 | X \gt 20] &= \frac{\operatorname{P}[X \leq 150] - \operatorname{P}[ X \leq 20]}{\operatorname{P}[X \leq 20]} \\ & = \frac{1-e^{-150/100}-1 + e^{-20/100}}{1-1 + e^{-20/100}} \\ &= \frac{-e^{-1.5} + e^{-0.2}}{e^{-0.2}} \\ &= 1-e^{-1.3} \\ &= 0.727. \end{align*} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.