Prescription drug losses, S, are modeled assuming the number of claims has a geometric distribution with mean 4, and the amount of each prescription is 40.

Calculate [math]\operatorname{E}[(S-100)_{+}][/math]

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

Key: C

Let N = number of prescriptions then

[math]n[/math]	[math]f_N(n)[/math]
0	0.2000
1	0.1600
2	0.1280
3	0.1024

[[math]] \begin{aligned} &\operatorname{E}[( S − 100) + ] = \operatorname{E}[ S ] − \operatorname{E}[ S \wedge 100] \\ &\operatorname{E}[ S ] = 40(4) = 160 \\ &\operatorname{E}[ S \wedge 100] = 0(0.2) + 40(0.16) + 80(0.128) + 100(1 − 0.2 − 0.16 − 0.128) = 67.84 \\ &\operatorname{E}[( S − 100) + ] = 160 − 67.84 = 92.16 \\ \end{aligned} [[/math]]

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