The unlimited severity distribution for claim amounts under an auto liability insurance policy is given by the cumulative distribution:

[[math]] F(x) = 1-0.8e^{-0.02x} - 0.2e^{-0.001x}, \, x \geq 0 [[/math]]

The insurance policy pays amounts up to a limit of 1000 per claim.

Calculate the expected payment under this policy for one claim.

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

Key: C

Limited expected value =

[[math]] \begin{aligned} &\int_{0}^{1000}[1 − F ( x)]dx = \int_{0}^{1000} 0.8e^{-0.02x} + 0.2e^{-0.001x}dx \\ &= -40e^{-0.02x} - 200e^{-0.001x} \Big |_0^{1000} \\ &= −0 − 73.576 + 40 + 200 \\ &= 166.424 \end{aligned} [[/math]]

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