Damages to a car in a crash are modeled by a random variable with density function

[[math]] f(x) = \begin{cases} c(x^2 − 60x + 800), \, 0 \lt x \lt 20 \\ 0, \, \textrm{otherwise} \end{cases} [[/math]]

where [math]c[/math] is a constant. A particular car is insured with a deductible of 2. This car was involved in a crash with resulting damages in excess of the deductible.

Calculate the probability that the damages exceeded 10.

• 0.12
• 0.16
• 0.20
• 0.26
• 0.78

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

Solution: D

[[math]] \begin{align*} 1 &= \int_{0}^{20} c(x^2-60x + 800) dx = c(x^3/3 - 30x^2 + 800x) \Big |_0^{20} = c20000/3 \Rightarrow c = 3/20000 \\ \operatorname{P}(X \gt d) &= \int_d^{20} c(x^2-60x + 800) dx = c(x^3/3 - 30x^2 + 800x) \Big |_d^{20} = 1- \frac{3}{20000}(d^3/3 -30d^2 + 800d) \\ \operatorname{P}(X\gt10 | X \gt2 ) &= \frac{\operatorname{P}(X\gt10)}{\operatorname{P}(X \gt 2)} = \frac{0.2}{0.776} = 0.2572. \end{align*} [[/math]]

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