Respuesta :
Answer:
The probability that, in the coming year, Company B’s total claim amount will exceed Company A’s total claim amount is 0.4013.
Step-by-step 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,
[tex]\mu_{A}=10,000\\\sigma_{A}=2,000[/tex]
The random variable B follows a Normal distribution with parameters,
[tex]\mu_{B}=9,000\\\sigma_{B}=2,000[/tex]
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:
[tex]E( A-B )=E(A)-E(B)=10000-9000=1000\\\\SD(A-B)=SD(A)+SD(B)-2Cov(A,B)=2000+2000-0=4000[/tex]
Compute the probability of A - B < 0 as follows:
[tex]P(A - B < 0)=P(\frac{(A-B)-E(A-B)}{SD(A-B)}<\frac{0-1000}{4000})[/tex]
            [tex]=P(Z<-0.25)\\=1-P(Z<0.25)\\=1-0.59871\\=0.40129\\\approx 0.4013[/tex]
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.
