You are given the following information about [math]N[/math], the annual number of claims for a randomly selected insured:

[[math]] \operatorname{P}[N = 0] = \frac{1}{2}, \, \operatorname{P}[N = 1] = \frac{1}{3}, \, \operatorname{P}[N \gt1] = \frac{1}{6} [[/math]]

Let [math]S[/math] denote the total annual claim amount for an insured. When [math]N = 1 [/math], [math]S[/math] is exponentially distributed with mean 5. When [math]N \gt 1 [/math], [math]S[/math] is exponentially distributed with mean 8.

Calculate [math]\operatorname{P}(4 \lt S \lt 8) [/math]

• 0.04
• 0.08
• 0.12
• 0.24
• 0.25

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

Solution: C

Observe

[[math]] \begin{align*} \operatorname{P}[ 4 \lt S \lt 8] &= \operatorname{P}[ 4 \lt S \lt 8 | N = 1] \operatorname{P}[ N = 1] + \operatorname{P}[ 4 \lt S \lt 8 | N \gt 1] \operatorname{P}[ N \gt 1] \\ &= \frac{1}{3}(e^{-4/5} - e^{-8/5}) + \frac{1}{6} \left( e^{-1/2} - e^{-1}\right) \\ &= 0.122. \end{align*} [[/math]]

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