Losses follow an exponential distribution with mean 1. Two independent losses are observed.

Calculate the expected value of the smaller loss.

• 0.25
• 0.50
• 0.75
• 1.00
• 1.50

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

Solution: B

Let X and Y be the two independent losses and Z = min(X,Y). Then,

[[math]] \operatorname{P}(Z \gt z) = \operatorname{P}(X \gt z \cap Y \gt z ) = \operatorname{P}(X \gt z ) \operatorname{P}(Y \gt z) = e^{-z}e^{-z} = e^{-2z} [[/math]]

[[math]] F_Z(z) = \operatorname{P}(Z \leq z) = 1-\operatorname{P}(Z \gt z) = 1-e^{-2z} [[/math]]

which can be recognized as an exponential distribution with mean 1/2.

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