You are given:

Number of Claims	Probability	Claim Size	Probability
0	1/5
1	3/5	25 150	1/3 2/3
2	1/5	50 200	2/3 1/3

Claim sizes are independent.

Calculate the variance of the aggregate loss.

• 4,050
• 8,100
• 10,500
• 12,510
• 15,612

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

Key: B

First obtain the distribution of aggregate losses:

Value	Probability
0	[math]1 / 5[/math]
25	[math](3 / 5)(1 / 3)=1 / 5[/math]
100	[math](1 / 5)(2 / 3)(2 / 3)=4 / 45[/math]
150	[math](3 / 5)(2 / 3)=2 / 5[/math]
250	[math](1 / 5)(2)(2 / 3)(1 / 3)=4 / 45[/math]
400	[math](1 / 5)(1 / 3)(1 / 3)=1 / 45[/math]

[[math]]\begin{aligned} & \mu=(1 / 5)(0)+(1 / 5)(25)+(4 / 45)(100)+(2 / 5)(150)+(4 / 45)(250)+(1 / 45)(400)=105 \\ & \sigma^{2}=(1 / 5)\left(0^{2}\right)+(1 / 5)(25)+(4 / 45)\left(100^{2}\right)+(2 / 5)\left(150^{2}\right)+(4 / 45)\left(250^{2}\right) \\ & +(1 / 45)\left(400^{2}\right)-105^{2}=8100 \end{aligned}[[/math]]

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