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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.

A solution has been posted here:

Expectation Poisson Distribution

However, I still don't understand why I can't just use the the 10000*{1 - P(X=0) - P(X=1)} + 0*{P(X=0) + P(X=1)}, where X is the number of major snowstorms a year to calculate the expectation of payment. It gives a different answer so it is not correct but I don't understand why. Could someone explain? Thanks!

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When you write $10000 \times (1 - P(X=0) - P(X=1))$ you are saying insurance pays $10000$ if $X \ge 2$, regardless of whether $X=2, 3,$ or $20$.

However, "The policy pays ... 10,000 for each one [after the first]" so the payment, when $X \ge 2$, is actually $10000 \times (X-1)$, instead of a constant value of $10000$.

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