An insurance policy reimburses a loss up to a benefit limit of 10. The policyholder’s loss, [math]Y[/math], follows a distribution with density function:

[[math]] f(y) = \begin{cases} 2y^{-3}, \, y \gt 1 \\ 0, \, \textrm{Otherwise.} \end{cases} [[/math]]

Calculate the expected value of the benefit paid under the insurance policy.

• 1.0
• 1.3
• 1.8
• 1.9
• 2.0

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

Solution: D

Let [math]W[/math] denote claim payments. Then

[[math]] W = \begin{cases} y, \quad 1 \lt y \lt 10 \\ 10, \quad y \leq 10 \end{cases} [[/math]]

It follows that

[[math]] \operatorname{E}[W] = \int_1^{10} y \frac{2}{y^3} dy + \int_{10}^{\infty} 10 \frac{2}{y^3} dy = -\frac{2}{y} \Big |_1^{10} - \frac{10}{y^2} \Big |_{10}^{\infty} = 1.9. [[/math]]

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