Question

Subject: Limit theorem

A small car insurance company has 10,000 policyholders. It was found that the expected annual claim per insured is (in dollars) of $ 240 with a deviation standard of $ 800. Estimate the likelihood that the company's total annual claim exceed $ 2,7 million.

Answer 1

The likelihood that total annual claims exceed 2.7million is the probability that Y is greater than 2.7 million.

Where Y = X1+X2 +......+ X(10,000)

As all Xi's are iid (here Xi is the annual claim per insured policymakers) and there are 10,000 policymakers)..

E(Xi) = 240 , std(Xi) = 800 ,so Var(Xi) =(800)^2 = 640000.

E(Y) = 10,000*240 = 2400000 = 2.4 million

And V(Y) = 10,000*640000 = 6400000000 ,hence std(Y) = √V(Y) = 80000 = 0.08 million..

Hence , we use central limit theorem so, (Y- E(Y))/std(Y) = z ~ N(0,1)

So, P(Y>2.7) = P( z > (2.7-2.4)/0.08) = P(z>3.75) = 0.00009

Hence , it's probability is very very rare event ( that occurs only 90 times in a million)...