Here is the question
A company buys a policy to insure its revenue in the event of major snow storms 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 snow storms 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'm pretty sure that I know how to solve most of the problem.
The payment about will look like this
0 x <=1
10,000(x-1) x > 1
The Poisson distribution will look like this
P(X=x)= e^1.5 (1.5^x/x!)
I know that the expected value should look like this
E[X] = P(X=0)(0) + P(X=1)(0) + P(X=2)(10,000) + P(X=3)(20,000) + ...
I know how to calculate the probability that X is greater than 1
P(X>1) = 1 - P(X=0) - P(X=1)
I'm just not sure how to incorporate the increasing payment sizes into the calculation.
Can somebody offer some guidance please? Does this have something to do with summing infinite sequences?