The number of policies that an agent sells has a Poisson distribution with modes at 2 and 3.

[math]K[/math] is the smallest number such that the probability of selling more than [math]K[/math] policies is less than 25%.

Calculate [math]K[/math].

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

Solution: D

Because the mode is 2 and 3, the parameter is 3 (when the parameter is a whole number the probabilities at the parameter and at one less than the parameter are always equal). Alternatively, the equation p(2) = p(3) can be solved for the parameter. Then the probability of selling 4 or fewer policies is 0.815 and this is the first such probability that exceeds 0.75. Thus, 4 is the first number for which the probability of selling more than that number of policies is less than 0.25.