The number of days that elapse between the beginning of a calendar year and the moment a high-risk driver is involved in an accident is exponentially distributed. An insurance company expects that 30% of high-risk drivers will be involved in an accident during the first 50 days of a calendar year.

Calculate the portion of high-risk drivers are expected to be involved in an accident during the first 80 days of a calendar year.

• 0.15
• 0.34
• 0.43
• 0.57
• 0.66

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

Solution: C

Let T denote the number of days that elapse before a high-risk driver is involved in an accident. Then T is exponentially distributed with unknown parameter λ . Now we are given that

[[math]] 0.3 = P[T ≤ 50] =50 \int_0^{50}\lambda e^{-\lambda t} \, dt = 1 - e^{-50\lambda}. [[/math]]

Therefore, [math]e^{–50\lambda} = 0.7 [/math] or [math]\lambda = − (1/50) \ln(0.7) [/math]. It follows that

[[math]] P[T ≤ 80] = \int_0^{80} \lambda e^{-\lambda t} \, dt = 1 - e^{-80 \lambda} = 1- e^{(80/50) \ln(0.7)} = 1-(0.7)^{80/50} = 0.435. [[/math]]

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