Calculate the expected amount paid

Question

The owner of a grocery shop buys a policy to insure its revenue in the event of catastrophes (e.g. earthquakes, snowstorms etc.) that shut down the business. 

The policy pays nothing for the first two catastrophe of the year and 50 000 for each one thereafter, until the end of the year.

The number of catastrophes per year that shut down the business is assumed to have a Poisson distribution with mean 3.

 

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

Answer 1

The probability of a single poisson event is e^-λλ^k/k!, where λ is the mean of the distribution (3 in this case).

The policy pays nothing on 0, 1, or 2 occurrences and pays 50,000 for each occurrence afterward. So we want the probability of 0, 1, or 2 occurrences and the probability of everything else or

P(N < 3) and

P(N > 2) = 1 - P(N < 3)

And our expected value is each probability multiplied by the corresponding payout:

E(X) = 0 * P(N < 3) + 50,000 * (1 - P(N < 3))

So we really only need P(N < 3). Use the formula for the poisson probability above.

P(N < 3) = P(N=0) + P(N=1) + P(N=2)

P(N = 0) = e^-33⁰/0! = e^-3

P(N = 1) = e^-33¹/1! = 3e^-3

P(N = 2) = e^-33²/2! = 9/2 * e^-3

So P(N < 3) = e^-3 + 3e^-3 + 9/2 * e^-3 = 17/2 * e^-3

Then plug into our expected value formula above:

E(X) = 0 * (17/2 * e^-3) + 50,000 * (1 - 17/2 * e^-3) ≈ $28,840.50