The annual profit of a life insurance company is normally distributed.

The probability that the annual profit does not exceed 2000 is 0.7642. The probability that the annual profit does not exceed 3000 is 0.9066.

Calculate the probability that the annual profit does not exceed 1000.

• 0.1424
• 0.3022
• 0.5478
• 0.6218
• 0.7257

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

Solution: C

[[math]] \begin{align*} \operatorname{P}(Z \leq 0.72) = 0.7642 = \operatorname{P}(X \leq 2000 ) = \operatorname{P}[Z \leq (2000-\mu)/\sigma] \\ 0.72 = (2000-\mu)/\sigma \\ \operatorname{P}( Z ≤ 1.32) = 0.9066 = \operatorname{P}( X ≤ 3000) = \operatorname{P}[ Z ≤ (3000 − µ ) / σ ] \\ 1.32 = (3000 − \mu ) / \sigma \\ 1.32 / 0.72 = (3000 − \mu ) / (2000 − \mu ) \\ 1.8333(2000 − \mu )= 3000 − \mu \\ \mu =[1.8333(2000) − 3000] / (1.8333 − 1) = 800 \\ \sigma =(3000 − \mu ) /1.32 = 1666.67 \\ \operatorname{P}( X ≤ 1000) = \operatorname{P}[ Z ≤ (1000 − 800) /1666.67] = \operatorname{P}( Z ≤ 0.12) = 0.5478. \end{align*} [[/math]]

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