Individuals purchase both collision and liability insurance on their automobiles. The value of the insured’s automobile is V. Assume the loss L on an automobile claim is a random variable with cumulative distribution function

[[math]] F(l) = \begin{cases} \frac{3}{4}\left(\frac{l}{V}\right)^3, \, 0 ≤ l \lt V \\ 1-\frac{1}{10}e^{\frac{-(l-V)}{V}}, \, \textrm{otherwise} \end{cases} [[/math]]

Calculate the probability that the loss on a randomly selected claim is greater than the value of the automobile.

• 0.00
• 0.10
• 0.25
• 0.75
• 0.90

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

Solution: B

[[math]] \operatorname{P}(X \gt V) = 1-\operatorname{P}( X \leq V ) = 1-F(V) = 1- (1-\frac{1}{10}e^{-\frac{V-V}{V}}) = 0.10 [[/math]]

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