A probability distribution of the claim sizes for an auto insurance policy is given in the table below:

Claim Size	Probability
20	0.15
30	0.10
40	0.05
50	0.20
60	0.10
70	0.10
80	0.30

Calculate the percentage of claims that are within one standard deviation of the mean claim size.

• 45%
• 55%
• 68%
• 85%
• 100%

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

Solution: A

Let X denote claim size. Then

E[X] = [20(0.15) + 30(0.10) + 40(0.05) + 50(0.20) + 60(0.10) + 70(0.10) + 80(0.30)] = (3 + 3 + 2 + 10 + 6 + 7 + 24) = 55

E[X2] = 400(0.15) + 900(0.10) + 1600(0.05) + 2500(0.20) + 3600(0.10) + 4900(0.10) + 6400(0.30) = 60 + 90 + 80 + 500 + 360 + 490 + 1920 = 3500

Var[X] = E[X2] – (E[X])2 = 3500 – 3025 = 475 and Var[ X ] = 21.79 .

Now the range of claims within one standard deviation of the mean is given by

[55.00 – 21.79, 55.00 + 21.79] = [33.21, 76.79]

Therefore, the proportion of claims within one standard deviation is

0.05 + 0.20 + 0.10 + 0.10 = 0.45 .

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