The number of claims [math]X[/math] on a health insurance policy is a random variable with [math]\operatorname{E}[ X^2 ] = 61[/math] and [math]\operatorname{E}[( X -1)^2 ] = 47 [/math] .

Calculate the standard deviation of the number of claims.

• 2.18
• 2.40
• 7.31
• 7.50
• 7.81

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

Solution: A

[[math]] \operatorname{E}[(X-1)^2] = \operatorname{E}[X^2] - 2\operatorname{E}[X] + 1 = 47 [[/math]]

so [math]\operatorname{E}[X] = (61 + 1 − 47) / 2= 7.5[/math]. The standard deviation is

[[math]] \sqrt{\operatorname{E}[X^2] -\operatorname{E}[X]^2} = \sqrt{61 - 7.5^2} = 2.18. [[/math]]

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