A company insures homes in three cities, J, K, and L. Since sufficient distance separates the cities, it is reasonable to assume that the losses occurring in these cities are mutually independent. The moment generating functions for the loss distributions of the cities are:

[[math]] M_J(t) = (1-2t)^{-3}, \, M_K(t) = (1-2t)^{-2.5}, \, M_L(t) = (1-2t)^{-4.5}. [[/math]]

Let [math]X[/math] represent the combined losses from the three cities. Calculate [math]\operatorname{E}[X^3][/math] .

• 1,320
• 2,082
• 5,760
• 8,000
• 10,560

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

Solution: E

Let [math]X_J, X_K,[/math] and [math]X_L[/math] represent annual losses for cities J, K, and L, respectively. Then [math]X = X_J + X_K + X_L[/math] and due to independence

[[math]] M(t) = M_J(t)M_K(t) M_L(t) = (1-2t)^{-3} (1-2t)^{-2.5} (1-2t)^{-4.5} = (1-2t)^{-10}. [[/math]]

Therefore,

[[math]] M'(t) = 20(1-2t)^{-11}, \, M^{''}(t) = 440(1-2t)^{-12}, \, M^{'''}(t) = 10560(1-2t)^{-13}, \, \operatorname{E}[X^3] = M^{'''}(0) = 10560. [[/math]]

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