Question

For Company A there is a 60% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean 10,000 and standard deviation 2,000. For Company B there is a 70% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean 9,000 and standard deviation 2,000. The total claim amounts of the two companies are independent. Calculate the probability that, in the coming year, Company B’s total claim amount will exceed Company A’s total claim amount.

Answer 1

The probability that, in the coming year, Company B’s total claim amount will exceed Company A’s total claim amount is 0.4013.

Explanation:

Let,

A = the total claim amount made for Company A during the coming year

B = the total claim amount made for Company B during the coming year

The random variable A follows a Normal distribution with parameters,

`\mu_(A)=10,000\\\sigma_(A)=2,000`

The random variable B follows a Normal distribution with parameters,

`\mu_(B)=9,000\\\sigma_(B)=2,000`

Compute the probability that in the coming year, Company B’s total claim amount will exceed Company A’s total claim amount as follows:

The variable is then: A - B < 0.

Compute the mean and standard deviation of A - B as follows:

`E( A-B )=E(A)-E(B)=10000-9000=1000\\\\SD(A-B)=SD(A)+SD(B)-2Cov(A,B)=2000+2000-0=4000`

Compute the probability of A - B < 0 as follows:

`P(A - B < 0)=P(((A-B)-E(A-B))/(SD(A-B))<(0-1000)/(4000))`

`=P(Z<-0.25)\\=1-P(Z<0.25)\\=1-0.59871\\=0.40129\\\approx 0.4013`

Thus, the probability that, in the coming year, Company B’s total claim amount will exceed Company A’s total claim amount is 0.4013.