Question

5. [15 masks] A satellite has a life expectancy that follows a normal distribution with μ = 96 months and ϭ = 6 months. An insurance company is going to insure the satellite for $500 million. a. If the satellite is insured for 90 months, what is the probability that it will stop functioning before the insurance coverage ends? b. If the insurance company charges $65 million for 90 months of insurance, how much profit does the insurance company expect to make (i.e., the expected value of the profit the insurance company will make)? c. For how many months should the satellite be insured to be 99% confident that it will last beyond the insurance date?

Answer 1

To find the probability that the satellite will stop functioning before the insurance coverage ends, calculate the z-score for the desired time frame and use a standard normal distribution table. To calculate the expected profit, subtract the insurance cost from the payout amount and multiply by the probability of the satellite failing within the coverage period. To determine the insurance coverage for 99% confidence, find the z-score corresponding to a cumulative probability of 0.99 and solve for the corresponding time frame.

Step-by-step explanation:

To answer question 5, we need to use the normal distribution. Given that the satellite has a life expectancy that follows a normal distribution with a mean (μ) of 96 months and a standard deviation (σ) of 6 months, we can calculate the probability that it will stop functioning before the insurance coverage ends.

a. Probability of satellite stopping functioning before 90 months:

To find this probability, we need to convert the time frame into a standardized z-score. Using the formula z = (x - μ) / σ, where x is the desired time frame:

z = (90 - 96) / 6 = -1

We can then use a standard normal distribution table or a calculator to find the area to the left of z = -1, which corresponds to the probability that the satellite will stop functioning before 90 months.

b. Expected profit for insurance company:

To find the expected profit, we need to subtract the insurance cost from the payout amount, and then multiply it by the probability of the satellite stopping functioning within the insurance coverage period.

Expected profit = (Payout amount - Insurance cost) * Probability of satellite stopping functioning within coverage period

c. Insurance coverage for 99% confidence:

To find the time frame for 99% confidence, we need to find the z-score that corresponds to a cumulative probability of 0.99. We can then use the formula z = (x - μ) / σ to find the corresponding time frame (x).

Z = 2.33 (approx)

2.33 = (x - 96) / 6

Solving for x, we find x = 97.98 (approx).