Posted by Wed Feb 14. Due: Wed Feb 21.
(Applying the concepts of a random variable and its mean.)
The Acme Shipping Company has learned from experience that it costs $14.80 to deliver a small package overnight. The company charges $20 for such a shipment, but guarantees that they will refund the $20 charge if it does not arrive within 24 hours.
Let X be a discrete random variable representing the outcomes for the Acme Shipping Company. What are the possible values for X?
Suppose Acme successfully delivers 96% of its packages within 24 hours. What are the probabilities that correspond to the values for X you found in the previous question?
Using the information from the previous two questions, what is the expected gain or loss for delivering a package?
Solve exercise 2.106 in the textbook.
Solve exercise 2.108 in the textbook.
Suppose that you work for an insurance company and you sell a $100,000 fire insurance policy at an annual premium of $1,350. Experience has shown that:
The probability of total loss (due to fire) to a house in that area and of the size of your customer’s house is .002 (in which case the insurance company will pay the full $100,000 to the customer).
The probability of 50% damage (due to fire) to a house in that area and of the size of your customer’s house is .008 (in which case the insurance company will pay only $50,000 to the customer).
For simplicity, we’ll ignore any other partial losses.
Let the random variable X be the insurance company’s annual gain from such a policy (i.e., the amount of money made by the insurance company from such a policy).
Find the probability distribution of X. In other words, list the possible values that X can have, and their corresponding probabilities. (Hint: There are three possibilities here: no fire, total loss due to fire, 50% damage due to fire).
What is the mean (expected) annual gain for a policy of this type? In other words, what is the mean of X?
The insurance company gets information about gas leakage in several houses that use the same gas provider that your customer does. In light of this new information, the probabilities of total loss and 50% damage (that were originally .002 and .008, respectively) are tripled (to .006 for total loss and .024 for 50% damage). Obviously, this change in the probabilities should be reflected in the annual premium, to account for the added risk that the insurance company is taking. What should be the new annual premium (instead of $1,350), if the company wants to keep its expected gain of $750? Guidance: Let the new premium (instead of 1,350) be denoted by N, for new. Set up the new probability distribution of X using the updated probabilities, and using N instead of 1,350. (The answer to question 1 will help.) The question now is: What should the value of N (the new premium) be, if we want the mean of X to remain 750? Set up an equation with N as unknown, and solve for N.
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