1. Loss amounts and claim numbers are independent within and between products.



2. 		Product X	Product Y
   Number of Claims	Mean	10	2
   	Standard Deviation	3	1
   Loss Amount	Mean	20	50
   	Standard Deviation	5	10




3. Aggregate losses, for both products combined, approximately follow the normal distribution.

Determine the probability that aggregate losses for both products combined exceed 400.

• Less than 0.10
• At least 0.10, but less than 0.15
• At least 0.15, but less than 0.20
• At least 0.20, but less than 0.25
• At least 0.25

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

Key: B

For Product X: aggregate losses have mean 10*20 = 200 and variance 10*25 + 400*9 = 3850.

For Product Y: aggregate losses have mean 2*50 = 100 and variance 2*100 + 2500 = 2700.

Because Product X and Product Y are independent, total aggregate losses, S, have mean 300 and variance 6550.

Using the normal approximation, we have:

[[math]] \operatorname{P}( S \gt 400) = \operatorname{P} \left( \frac{S-300}{\sqrt{6550}} \gt \frac{400-300}{\sqrt{6550}} \right)= \operatorname{P}(Z \gt 1.24 ) = 0.11 [[/math]]

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