A government employee’s yearly dental expense follows a uniform distribution on the interval from 200 to 1200. The government’s primary dental plan reimburses an employee for up to 400 of dental expense incurred in a year, while a supplemental plan pays up to 500 of any remaining dental expense. Let [math]Y[/math] represent the yearly benefit paid by the supplemental plan to a government employee.

Calculate [math]\operatorname{Var}(Y)[/math].

• 20,833
• 26,042
• 41,042
• 53,333
• 83,333

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

Solution: C

Let X represent individual expense. Then,

[[math]] Y = \begin{cases} 0, \, 200 \leq X \leq 400 \\ X-400, \, 400 \lt X \lt 900 \quad \textrm{and the density function of X is} \, f(x) = 0.001, \, 200 \leq x \leq 1200 \\ 500, \, 900 \lt X \leq 1200 \end{cases} [[/math]]

[[math]] \begin{align*} \operatorname{E}(Y) &= \int_{200}^{400} 0(0.001) dx + \int_{400}^{900} (x-400)(0.001) dx + \int_{900}^{1200} 500(0.001) dx \\ &= 0 + 0.001 \frac{(x-400)^2}{2} \Big |_{400}^{900} + 500(0.001)(1200-900) \\ &= 0 + 125 + 150 = 275 \end{align*} [[/math]]

[[math]] \begin{align*} \operatorname{E}(Y^2) &= \int_{200}^{400} 0^2(0.001) dx + \int_{400}^{900} (x-400)^2(0.001) dx + \int_{900}^{1200} 500^2(0.001) dx \\ &= 0 + 0.001 \frac{(x-400)^3}{3} \Big |_{400}^{900} + 500^2(0.001)(1200-900) \\ &= 0 + 41, 666.67 + 75, 000 = 116, 666.67 \end{align*} [[/math]]

[[math]] \operatorname{Var}(Y) = 116, 666.67 − 275^2 = 41, 041.67. [[/math]]

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