Chapter 6: Random Variables
Life insurance A life insurance company sells a term insurance policy to a-year-old male that pays if the insured dies within the next years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of each year as payment for the insurance. The amount Y that the company earns on this policy is per year, less the that it must pay if the insured dies. Here is a partially completed table that shows information about risk of mortality and the values of profit earned by the company:
(a) Copy the table onto your paper. Fill in the missing values of .
(b) Find the missing probability. Show your work.
(c) Calculate the mean . Interpret this value in context.
Compute and interpret the standard deviation of
Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of 1 (low) to 5 (high). Let X be the pain score for a randomly selected patient. The following table gives part of the probability distribution for X.
(b) If two patients who received artificial knees are chosen at random, what’s the probability that both of them report pain scores of or ? Show your work.
(c) Compute the mean and standard deviation of . Show your work.
Toss times Suppose you toss a fair coin times. Let the number of heads you get.
(a) Find the probability distribution of.
(b) Make a histogram of the probability distribution. Describe what you see.
(c) Find and interpret the result.
What is the probability that a randomly chosen subject completes at least puzzles in the five-minute period while listening to soothing music?
(a) 0.3 (c) 0.6 (e) Cannot be determined
(b) 0.4 (d) 0.9
Fire insurance Suppose a homeowner spends for a home insurance policy that will pay out if the home is destroyed by fire. Let the profit made by the company on a single policy. From previous data, the probability that a home in this area will be destroyed by fire is .
(a) Make a table that shows the probability distribution of Y.
(b) Compute the expected value of Y. Explain what this result means for the insurance company
A test for extrasensory perception (ESP) involves asking a person to tell which of shapes—a circle, star, triangle, diamond, or heart—appears on a hidden computer screen. On each trial, the computer is equally likely to select any of the shapes. Suppose researchers are testing a person who does not have ESP and so is just guessing on each trial. What is the probability that the person guesses the first shapes incorrectly but gets the fifth correct?
Using Benford's law According to Benford's law (Exercise , page ), the probability that the first digit of the amount of a randomly chosen invoice is an or a is . Suppose you examine randomly selected invoices from a vendor until you find one whose amount begins with an or a .
(a) How many invoices do you expect to examine until you get one that begins with an or ? Justify your answer.
(b) In fact, you don't get an amount starting with an or until the invoice. Do you suspect that the invoice amounts are not genuine? Compute an appropriate probability to support your answer.
101. Job reads that 1 out of 4 eggs contains salmonella bacteria. So he never uses more than 3 eggs in cooking. If eggs do or don't contain salmonella independently of each other, the number of contaminated eggs when Joe uses 3 chosen at random has the following distribution:
(a) binomial; and
(b) binomial; and
(c) binomial; and
In the previous exercise, the probability that at least of Joe's eggs contains salmonella is about