The number of claims received in a year by an insurance comp

The number of claims received in a year by an insurance company is a Poisson random variable with lambda = 500. The claim amounts are independent and uniformly distributed over [0, 500]. The total claim amount is a random variable S. The total claim amount this year is one standard deviation above the mean (it means the total claim amount is E(S) + square root V(S)). If the company has $140,000 available, will it be enough to pay all the claims filed this year?

Solution

Claim is a uniform variable

Hence average of S = 250

Variance of S = (500-0)^2/12 = 20833.33

Std dev =144.3376

E(s)+std dev (s) = 394.3376

No of claims that can be settled with 140000=355.0257

No of claims follow Poisson with average 500

Hence amount will not be sufficient to meet the claims of average number of 500