In a small metropolitan area, annual losses due to storm, fire, and theft are assumed to be mutually independent, exponentially distributed random variables with respective means 1.0, 1.5, and 2.4.

Calculate the probability that the maximum of these losses exceeds 3.

• 0.002
• 0.050
• 0.159
• 0.287
• 0.414

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

Solution: E

Let [math]S, F,[/math] and [math]T[/math] be the losses due to storm, fire, and theft respectively. Let [math]Y = \max(S,F,T)[/math]. Then,

[[math]] \begin{align*} \operatorname{P}[Y \gt 3] =1 − \operatorname{P}[Y ≤ 3] =1 − \operatorname{P}[\max( S , F , T ) ≤ 3] &=1 − \operatorname{P}[ S ≤ 3]\operatorname{P}[ F ≤ 3]\operatorname{P}[T ≤ 3]\\ &= 1 − (1 − e^{−3/1} )(1 − e^{−3/1.5} )(1 − e ^{-3/2.4} ) \\ &=0.414. \end{align*} [[/math]]

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