An insurance company has an equal number of claims in each of three territories. In each territory, only three claim amounts are possible: 100, 500, and 1000. Based on the company’s data, the probabilities of each claim amount are:

	Total Amount
	100	500	1000
Territory 1	0.90	0.08	0.02
Territory 2	0.80	0.11	0.09
Territory 3	0.70	0.20	0.10

Calculate the standard deviation of a randomly selected claim amount.

• 254
• 291
• 332
• 368
• 396

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

Solution: A

Because the territories are evenly distributed, the probabilities can be averaged. Thus the probability of a 100 claim is 0.80, of a 500 claim is 0.13, and of a 1000 claim as 0.07. The mean is 0.80(100) + 0.13(500) + 0.07(1000) = 215. The second moment is 0.80(10,000) + 0.13(250,000) + 0.07(1,000,000) = 110,500. The variance is 110,500 – (215)(215) = 64,275. The standard deviation is 253.53.

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