Expectation Poisson Distribution 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. What is the expected amount paid to the company under this policy during a one-year period?
I know how to calculate the expectation and what the series is. I'm having problems with the summations. I know it should involve:
$$\sum_{k=2}^{+\infty} \frac{(1.5)^k}{k!}$$
 A: Let $X$ be the number of snowstorms occurring in the given year and let $Y$ be the amount paid to the company.  Call one unit of money $\$ 10{,}000$.
Then $Y$ takes the value $0$ when $X=0$ or $X=1$, the value  $1$ when $X=2$, the value $2$ when $X=3$, etc..
The expected payment is
$$\eqalign{
\Bbb E(Y)
&=\sum_{k=2}^\infty (k-1)P[X=k]\cr
&=\sum_{k=2}^\infty (k-1) e^{-1.5}{(1.5)^k\over k!}\cr
&=\sum_{k=1}^\infty (k-1) e^{-1.5}{(1.5)^k\over k!}\cr
&=
\sum_{k=1}^\infty k e^{-1.5}{(1.5)^k\over k!}
-\sum_{k=1}^\infty  e^{-1.5}{(1.5)^k\over k!}\cr
&=\underbrace{ \sum_{k=0}^\infty k e^{-1.5}{(1.5)^k\over k!}}_{\text{mean of } X} -
\biggl(-e^{-1.5}+\underbrace{\sum_{k=0}^\infty  e^{-1.5}{(1.5)^k\over k!}}_{=1}\biggr)\cr
&=1.5+e^{-1.5}- 1\cr
&=0.5+e^{-1.5}\cr
&\approx .7231\,\text{units}.
}$$
A: The mean of the plain Poisson is $1.5$. So if it paid for all snowstorms, the mean outlay would be $15000$.
The probability of one storm is $1.5e^{-1.5}$. The company does not pay for this, so subtract $15000e^{-1.5}$ from $15000$.
A: The expected payout is actually
$$\begin{align} 10,000 \sum_{k=2}^{\infty} (k-1) \frac{(1.5)^k}{k!} e^{-1.5} &= 10,000 (1.5) \sum_{k=1}^{\infty} \frac{(1.5)^k}{k!} e^{-1.5} - 10,000 \sum_{k=2}^{\infty} \frac{(1.5)^k}{k!} e^{-1.5}  \\ &= 10,000 (1.5) (1-e^{-1.5}) - 10,000 (1-2.5 e^{-1.5})\\ &\approx 7231.30 \\ \end{align} $$
A: If they got $\$10,000$ every time, it would be $\$10,000\cdot(1.5)$.  From that subtract $\$10,000$ times the probability that there's exactly one such storm, which is $1.5e^{-1.5}$.  So you get
$$
\$15,000 - \$10,000\cdot1.5e^{-1.5}.
$$
