An actuary is studying hurricane models. A year is classified as a high, medium, or low hurricane year with probabilities 0.1, 0.3, and 0.6, respectively. The numbers of hurricanes in high, medium, and low years follow Poisson distributions with means 20, 15, and 10, respectively.

Calculate the variance of the number of hurricanes in a randomly selected year.

• 11.25
• 12.50
• 12.94
• 13.42
• 23.75

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

Solution: E

The mean is the weighted average of the three means: 0.1(20) + 0.3(15) + 0.6(10) = 12.5. The second moment is the weighted average of the three second moments (each of which is the square of the mean plus the mean, for a Poisson distribution): 0.1(420) + 0.3(240) + 0.6(110) = 180. The variance is the second moment minus the square of the mean, which is 23.75.

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