SOLUTION: A workplace injury policy has a premium (policy cost) of $7. The probability of a claim is 0.43, in which case the amount of the claim has density 1.2/(𝑥^2.2), when 𝑥 ≥ 1.

Algebra ->  Probability-and-statistics -> SOLUTION: A workplace injury policy has a premium (policy cost) of $7. The probability of a claim is 0.43, in which case the amount of the claim has density 1.2/(𝑥^2.2), when 𝑥 ≥ 1.      Log On


   

Question 1200583: A workplace injury policy has a premium (policy cost) of $7. The probability of a claim is
0.43, in which case the amount of the claim has density 1.2/(𝑥^2.2), when 𝑥 ≥ 1. Find the expected
value of the net revenue to the company.
Answer by GingerAle(43) About Me  (Show Source):
You can put this solution on YOUR website!
**1. Define Net Revenue**
* **Net Revenue = Premium Revenue - Expected Claim Cost**
**2. Calculate Premium Revenue**
* Premium Revenue = Premium per policy * Probability of no claim + Premium per policy * Probability of claim
* Premium Revenue = $7 * (1 - 0.43) + $7 * 0.43 = $7
**3. Calculate Expected Claim Cost**
* **Expected Claim Cost = Probability of claim * Expected claim amount**
* **Find Expected Claim Amount:**
* Expected Claim Amount = ∫(x * f(x)) dx
where f(x) is the probability density function of the claim amount
and the integral is taken over the range of possible claim amounts (x ≥ 1)
* Expected Claim Amount = ∫(x * (1.2/x^2.2)) dx from 1 to infinity
* Expected Claim Amount = 1.2 ∫(x^(-1.2)) dx from 1 to infinity
* To solve this improper integral, we need to find the limit as the upper bound approaches infinity:
* Expected Claim Amount = 1.2 * lim (b→∞) [(-1/0.2) * x^(-0.2)] from 1 to b
* Expected Claim Amount = -6 * lim (b→∞) [b^(-0.2) - 1^(-0.2)]
* Expected Claim Amount = -6 * (0 - 1) = 6
* **Expected Claim Cost = 0.43 * 6 = 2.58**
**4. Calculate Net Revenue**
* Net Revenue = $7 - $2.58 = $4.42
**Therefore, the expected value of the net revenue to the company is $4.42.**