An airport purchases an insurance policy to offset costs associated with excessive amounts of snowfall. For every full ten inches of snow in excess of 40 inches during the winter season, the insurer pays the airport 300 up to a policy maximum of 700. The following table shows the probability function for the random variable [math]X[/math] of annual (winter season) snowfall, in inches, at the airport.

Inches	[0,20)	[20,30)	[30,40)	[40,50)	[50,60)	[60,70)	[70,80)	[80,90)	[90,inf)
Probability	0.06	0.18	0.26	0.22	0.14	0.06	0.04	0.04	0.00

Calculate the standard deviation of the amount paid under the policy.

• 134
• 235
• 271
• 313
• 352

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

Solution: B

The payments are 0 with probability 0.72 (snowfall up to 50 inches), 300 with probability 0.14, 600 with probability 0.06, and 700 with probability 0.08. The mean is 0.72(0) + 0.14(300) + 0.06(600) + 0.08(700) = 134 and the second moment is 0.72(0²) + 0.14(300²) + 0.06(600²) + 0.08(700²) = 73,400. The variance is 73,400 – 134² = 55,444. The standard deviation is the square root, 235.

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