Losses due to accidents at an amusement park are exponentially distributed. An insurance company offers the park owner two different policies, with different premiums, to insure against losses due to accidents at the park.

Policy A has a deductible of 1.44. For a random loss, the probability is 0.640 that under this policy, the insurer will pay some money to the park owner.

Policy B has a deductible of d. For a random loss, the probability is 0.512 that under this policy, the insurer will pay some money to the park owner.

Calculate d.

• 0.960
• 1.152
• 1.728
• 1.800
• 2.160

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

Solution: E

For Policy A, the relevant equation is

[[math]] \begin{align*} 0.64 &=P( L \gt 1.44) = e^{-1.44/\mu} \\ \ln(0.64) &= -0.44629 = -1.44/\mu \\ \mu &= 3.2266. \end{align*} [[/math]]

For Policy B, the relevant equation is

[[math]] \begin{align*} 0.512 &=P( L \gt d) = e^{-d/3.2266} \\ \ln(0.512) &= -0.6694 = -d/3.2266 \\ d &= 2.1599. \end{align*} [[/math]]

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