Claim amounts are independent random variables with probability density function

[[math]] f(x) = \begin{cases} \frac{10}{x^2}, \, x \gt10 \\ 0, \, \textrm{otherwise} \end{cases} [[/math]]

Calculate the probability that the largest of three randomly selected claims is less than 25.

• 8/125
• 12/125
• 27/125
• 2/5
• 3/5

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

Solution: C

The probability that a single claim is less than 25 is

[[math]] \operatorname{P}( X \leq 25) = \int_{10}^{25} 10x^{-2} dx = 3/5. [[/math]]

The probability that all three claims are less than 25 is (3 / 5)³ = 27 /125.

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