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Chapter 6: Q.4 (page 409)

A certain vending machine offers 20-ounce bottles of soda for $1.50. The number of bottles X bought from the machine on any day is a random variable with mean 50 and standard deviation 15. Let the random variable Y equal the total revenue from this machine on a given day. Assume that the machine works properly and that no sodas are stolen from the machine. What are the mean and standard deviation of Y?

Short Answer

Expert verified

The mean and standard deviation of Y is μY = $75, σY = $22.50

Step by step solution

01

Given information

μx = 50

σx = 15

02

Explanation

Y = 1.50X

μaX+b = aμX+b

σaX+b = aσX

μY =μ1.5X = 1.5μ

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Most popular questions from this chapter

49. Checking independence In which of the following games of chance would you be willing to assume independence of Xandlocalid="1649903939419" Y in making a probability model? Explain your answer in each case.
(a) In blackjack, you are dealt two cards and examine the total points localid="1649903945891" Xon the cards (face cards count localid="1649903950298" 10points). You can choose to be dealt another card and compete based on the total pointslocalid="1649903956461" Y on all three cards.
(b) In craps, the betting is based on successive rolls of two dice. localid="1649903966379" Xis the sum of the faces on the first roll, and localid="1649903971075" Yis the sum of the faces on the next roll.

51. His and her earnings A study of working couples measures the income Xof the husband and the income Yof the wife in a large number of couples in which both partners are employed. Suppose that you knew the means μXand μYand the variances σX2and σγ2 of both variables in the population.
(a) Is it reasonable to take the mean of the total income X+Yto be μX+μY ? Explain your answer.
(b) Is it reasonable to take the variance of the total income to be σX2+σy2 ? Explain your answer.

Chess and reading (4.3) If the study found a statistically significant improvement in reading scores, could you conclude that playing chess causes an increase in reading skills? Justify your answer

18. Life insurance

(a) It would be quite risky for you to insure the life of a 21-year-old friend under the terms of Exercise 14. There is a high probability that your friend would live and you would gain \(1250in premiums. But if he were to die, you would lose almost \)100,000. Explain carefully why selling insurance is not risky for an insurance company that insures many thousands of 21-year-old men.

(b) The risk of an investment is often measured by the standard deviation of the return on the investment. The more variable the return is, the riskier the
investment. We can measure the great risk of insuring a single person’s life in Exercise 14by computing the standard deviation of the income Y that the insurer will receive. Find σY using the distribution and mean found in Exercise 14.

Refer to the previous Check Your Understanding (page 390 ) about Mrs. Desai's special multiple-choice quiz on binomial distributions. We defined X=the number of Patti's correct guesses.

1. Find μX. Interpret this value in context.
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