Life Insurance Project, Part 1: Functions of random variables are used as mathematical

Chapter 9, Problem 9

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Life Insurance Project, Part 1: Functions of random variables are used as mathematical models in the insurance business. The following numbers were taken from a mortality table. The table shows the probability, P(x), that a person who is alive on his or her xth birthday will die before he or she reaches age x + 1. A group of 10,000 15-year-olds get together to form their own life insurance company. For a premium of $40 per year, they agree to pay $20,000 to the family of anyone in the group who dies while he or she is 15 through 20 years old. Age, x P(x) 15 0.00146 16 0.00154 17 0.00162 18 0.00169 19 0.00174 20 0.00179 a. Calculate D(15), the number out of 10,000 expected to die while they are 15. Round to an integer. b. Calculate A(16), the number out of 10,000 expected to be alive on their 16th birthday. c. Calculate D(16). Round to an integer. d. Make a table of x, P(x), A(x), and D(x) for each value of x from 15 through 20. e. Put columns in the table of part d for I(x) and O(x), the income from the $40 premiums and the amount paid out from the $20,000 death benefits. Take into account that a person who dies no longer pays premiums the following years. f. Calculate NI(x) = I(x) O(x), the net income of the company each year. Explain why NI(x) decreases each year. (There are reasons!) g. On average, how much would the company expect to make per year? Would this be enough to pay a full-time employee to operate the company?