A company buys a policy to insure its revenue in the event of major snowstorms that shut down business. The policy pays nothing for the first such snowstorm of the year and 10,000 for each one thereafter, until the end of the year. The number of major snowstorms per year that shut down business is assumed to have a Poisson distribution with mean 1.5.

Calculate the expected amount paid to the company under this policy during a one-year period.

• 2,769
• 5,000
• 7,231
• 8,347
• 10,578

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

Solution: C

Let [math]N[/math] be the number of major snowstorms per year, and let [math]P[/math] be the amount paid to to the company under the policy. Then

[[math]] \operatorname{P}[N = n] = \frac{(3/2)^ne^{-3/2}}{n!}, n = 0, 1, 2, \ldots [[/math]]

and

[[math]] P = \begin{cases} 0, \quad N =0 \\ 10000(N-1), \quad N \geq 1 \end{cases} [[/math]]

Now observe that

[[math]] \begin{align*} \operatorname{E}[P] &= \sum_{n=1}^{\infty} 10000(n-1) \frac{(3/2)^ne^{-3/2}}{n!} \\ &= 10000 e^{-3/2} + \sum_{n=0}^{\infty} 10000(n-1) \frac{(3/2)^{n}e^{-3/2}}{n!} \\ &= 10000 e^{-3/2} + \operatorname{E}[10000(N-1)] \\ &= 10000 e^{-3/2} + \operatorname{E}[10000N] - \operatorname{E}[10000] \\ & = 10000e^{-3/2} + 10000 (3/2) - 10000 \\ &= 7231. \end{align*} [[/math]]

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