A loss under a liability policy is modeled by an exponential distribution. The insurance company will cover the amount of that loss in excess of a deductible of 2000. The probability that the reimbursement is less than 6000, given that the loss exceeds the deductible, is 0.50. Calculate the probability that the reimbursement is greater than 3000 but less than 9000, given that the loss exceeds the deductible.

• 0.28
• 0.35
• 0.50
• 0.65
• 0.72

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

Solution: B

Due to the memoryless property of the exponential distribution, the distribution of the reimbursement given that there is a payment is exponential with the same parameter. Thus

[[math]] 0.5 = F(6000) = 1-e^{-6000/2} [[/math]]

which implies that [math]\lambda = 8656.17 [/math]. The solution is

[[math]] F(9000) - F(3000) = (1-e^{-9000/8656.17})-(1-e^{-3000/8656.17}) = 0.35. [[/math]]

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