The number of claims, N, made on an insurance portfolio follows the following distribution:

[math]n[/math]	[math]\operatorname{Pr}(N=n)[/math]
0	0.7
2	0.2
3	0.1

If a claim occurs, the benefit is 0 or 10 with probability 0.8 and 0.2, respectively.

The number of claims and the benefit for each claim are independent.

Calculate the probability that aggregate benefits will exceed expected benefits by more than 2 standard deviations.

• 0.02
• 0.05
• 0.07
• 0.09
• 0.12

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

Key: E

[[math]] \begin{aligned} &\operatorname{E}[ N ] = 0.7(0) + 0.2(2) + 0.1(3) = 0.7 \\ &\operatorname{E}[ N ] = 0.7(0) + 0.2(4) + 0.1(9) − 0.7 2 = 1.21 \\ &\operatorname{E}[ X ] = 0.8(0) + 0.2(10) = 2 \\ &\operatorname{E}[ X ] = 0.8(0) + 0.2(100) − 2 2 = 16 \\ &\operatorname{E}[ S ] = \operatorname{E}[ N ] \operatorname{E}[ X ] = 0.7(2) = 1.4 \\ &\operatorname{E}[ S ] = \operatorname{E}[ N ]\operatorname{E}[ X ] + \operatorname{E}[ X ]^2\operatorname{E}[ N ] = 0.7(16) + 4(1.21) = 16.04 \\ &\operatorname{SD}( S ) = 16.04 = 4 \\ &\operatorname{Pr}( S \gt 1.4 + 2(4) = 9.4) = 1 − \operatorname{Pr}( S = 0) = 1 − 0.7 − 0.2(0.8)^2 − 0.1(0.8)^3 = 0.12 \end{aligned} [[/math]]

The last line follows because there are no possible values for S between 0 and 10. A value of 0 can be obtained three ways: no claims, two claims both for 0, three claims all for 0.

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