An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. The number of claims filed has a Poisson distribution.

Calculate the variance of the number of claims filed.

• [math]\frac{1}{\sqrt{3}}[/math]
• 1
• [math]\sqrt{2}[/math]
• 2
• 4

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

Solution: D

Let [math]N[/math] be the number of claims filed. We are given

[[math]] P[N = 2] = \frac{e^{-\lambda}\lambda^2}{2!} = 3 \frac{e^{-\lambda}\lambda^4}{4!} = 3 \cdot P[N=4] 24 \lambda^2 = 6 \lambda^4 [[/math]]

which implies that [math]\lambda = 2 [/math]. Therefore, [math]\operatorname{Var}[N] = \lambda = 2 [/math].

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