An insurance policy reimburses dental expense, [math]X[/math], up to a maximum benefit of 250. The probability density function for [math]X[/math] is:

[[math]] f(x) = \begin{cases} ce^{-0.004x}, \, x \gt 0 \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]

where [math]c[/math] is a constant. Calculate the median benefit for this policy.

• 161
• 165
• 173
• 182
• 250

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

Solution: C

Note that [math]X[/math] has an exponential distribution. Therefore, c = 0.004 . Now let [math]Y[/math] denote the claim benefits paid. Then

[[math]] Y = \begin{cases} x, \quad x \leq 250 \\ 250, \quad x \geq 250 \end{cases} [[/math]]

and we want to find [math]m[/math] such that 0.50 equals

[[math]] \int_0^{\infty} 0.004e^{-0.004x} dx = -e^{-0.004x} \Big |_0^m = 1- e^{-0.004m}. [[/math]]

This condition implies

[[math]] e^{-0.004m} = 0.5 \Rightarrow m = 250 \ln(2) = 173.29. [[/math]]

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