The number of days an employee is sick each month is modeled by a Poisson distribution with mean 1. The numbers of sick days in different months are mutually independent.

Calculate the probability that an employee is sick more than two days in a three-month period.

• 0.199
• 0.224
• 0.423
• 0.577
• 0.801

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

Solution: D

Let N be the number of sick days for an employee in three months. The sum of independent Poisson variables is also Poisson and thus N is Poisson with a mean of 3. Then,

[[math]] \operatorname{P}[ N \leq 2 ] = e^{-3} \left ( \frac{3^0}{0!} + \frac{3^1}{1!} + \frac{3^2}{2!} \right ) = e^{-3}(1 + 3 + 4.5) = 0.423. [[/math]]

The answer is complement, 1 - 0.423 = 0.577.

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