A probability distribution of the claim sizes for an auto in

A probability distribution of the claim sizes for an auto insurance policy is given in the table below: Left column: Claim Size; right column: probability of occurrence of this claim. What percentage of claims are within one standard deviation of the mean claim size?

20 - 0.15

30 - 0.1

40 - 0.05

50 - 0.20

60 - 0.10

70 - 0.10

80 - 0.30

Solution

As

Mean = Sum (x P(x))

and

standard deviation = sqrt[ Sum(x^2 P(x)) - (Sum (x P(x)))^2]

Then

x   P(x)   x P(x)   x^2 P(x)
20   0.15   3   60
30   0.1   3   90
40   0.05   2   80
50   0.2   10   500
60   0.1   6   360
70   0.1   7   490
80   0.3   24   1920

Sum( xP(x)) = 55
Sum(x^2 P(x)) = 3500

Thus,

Mean = 55
s = 21.79449472

Thus, the interval is from (33.20550528 to 76.79449472).

Thus, within one standard deviation is 40 to 70.

Adding P(40) + P(50) + P(60) + P(70),

P(within 1 standard deviation) = 0.45 [ANSWER]