Question

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.
1. What is the expected amount paid to the company under this policy during a one-year period?

Answer 1

Answer:

Step-by-step explanation:

Let X be the number of snowstorms

let Y be the  paid amount

Call one unit of money 10,000

If X = 0 or 1 then y = 0

X = 2 then y = 1 so on ...

E(Y) =  $\sum_{k=2}^{\infty}(k-1)P[X=k]$

=  $\sum_{k=2}^{\infty}(k-1)e^{^{-1.5}}\frac{(1.5)^{k}}{k!}$

=  $\sum_{k=1}^{\infty}(k-1)e^{^{-1.5}}\frac{(1.5)^{k}}{k!}$

=  $\sum_{k=1}^{\infty}ke^{^{-1.5}}\frac{(1.5)^{k}}{k!}$ - $\sum_{k=1}^{\infty}e^{^{-1.5}}\frac{(1.5)^{k}}{k!}$

=  $\sum_{k=0}^{\infty}ke^{^{-1.5}}\frac{(1.5)^{k}}{k!}$-(-$e^{-1.5}$ +$\sum_{k=0}^{\infty}e^{^{-1.5}}\frac{(1.5)^{k}}{k!}$ )

= 1.5 +$e^{-1.5}$ - 1 ≈ 0.7231