The number of hurricanes that will hit a certain house in the next ten years is Poisson distributed with mean 4. Each hurricane results in a loss that is exponentially distributed with mean 1000. Losses are mutually independent and independent of the number of hurricanes.

Calculate the variance of the total loss due to hurricanes hitting this house in the next ten years.

• 4,000,000
• 4,004,000
• 8,000,000
• 16,000,000
• 20,000,000

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

Solution: C

Let [math]N[/math] denote the number of hurricanes, which is Poisson distributed with mean and variance 4.

Let [math]X_i[/math] denote the loss due to the [math]i^{th}[/math] hurricane, which is exponentially distributed with mean 1,000 and therefore variance 1,000² = 1,000,000.

Let [math]X[/math] denote the total loss due to the [math]N[/math] hurricanes.

This problem can be solved using the conditional variance formula. Note that independence is used to write the variance of a sum as the sum of the variances.

[[math]] \begin{align*} \operatorname{Var}(X) &= \operatorname{Var}[ \operatorname{E}( X | N )] + E[\operatorname{Var}( X | N )] \\ &= \operatorname{Var}[ \operatorname{E}( X_1 + \cdots + X_N )] + E[\operatorname{Var}( X_1 + \cdots + X_N )] \\ &= \operatorname{Var}(1, 000 N ) + \operatorname{E}(1, 000, 000 N ) \\ &= 1, 000 ^ 2\operatorname{Var}( N ) + 1, 000, 000 \operatorname{E}( N ) \\ &= 1, 000, 000(4) + 1, 000, 000(4) = 8, 000, 000. \end{align*} [[/math]]

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