Tan S.T. Finite Mathematics for the Managerial, Life, and Social Sciences

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Solution

Let X be a random variable whose values are 1 and 1, which corre-

spond to a win and a loss. The probabilities associated with the values 1 and 1

are and , respectively. Therefore, the expected value is given by

Thus, if one places a $1 bet on red over and over again, one may expect to lose,

on the average, approximately 5 cents per bet in the long run.

Examples 5 and 6 illustrate games that are not “fair.” Of course, most participants in

such games are aware of this fact and participate in them for other reasons. In a fair

game, neither party has an advantage, a condition that translates into the condition that

E(X)  0, where X takes on the values of a player’s winnings.

APPLIED EXAMPLE 7

Fair Games Mike and Bill play a card game

with a standard deck of 52 cards. Mike selects a card from a well-shuffled

deck and receives A dollars from Bill if the card selected is a diamond; otherwise,

Mike pays Bill a dollar. Determine the value of A if the game is to be fair.

Solution

Let X denote a random variable whose values are associated with

Mike’s winnings. Then X takes on the value A with probability P(X  A) 

(since there are 13 diamonds in the deck) if Mike wins and takes on the value 1

with probability P(X 1)  if Mike loses. Since the game is to be a fair one,

the expected value E(X) of Mike’s winnings must be equal to zero; that is,

Solving this equation for A gives A  3. Thus, the card game will be fair if Bill

makes a $3 payoff when a diamond is drawn.

Odds

In everyday parlance, the probability of the occurrence of an event is often stated

in terms of the odds in favor of (or odds against) the occurrence of the event. For

example, one often hears statements such as “the odds that the Dodgers will win the

World Series this season are 7 to 5” and “the odds that it will not rain tomorrow are 3

to 2.” We will return to these examples later. But first, let us look at a definition that

ties together these two concepts.

E1X2 A a

b 112a

b 0

⬇ 0.053

E1X 2 1 a

b 112a

b

432 8 PROBABILITY DISTRIBUTIONS AND STATISTICS

Odds In Favor Of and Odds Against

If P(E) is the probability of an event E occurring, then

1. The odds in favor of E occurring are

(3a)

2. The odds against E occurring are

(3b)

1  P1E2

P1E2



P1E

P1E2

3P1E2 04

P1E2

1  P1E2



P1E2

P1E

3P1E2 14

87533_08_ch8_p417-482 1/30/08 10:09 AM Page 432

Notes

1. The odds in favor of the occurrence of an event are given by the ratio of the prob-

ability of the event occurring to the probability of the event not occurring. The odds

against the occurrence of an event are given by the reciprocal of the odds in favor

of the occurrence of the event.

2. Whenever possible, odds are expressed as ratios of whole numbers. If the odds in

favor of E are , we say the odds in favor of E are a to b. If the odds against E occur-

ring are , we say the odds against E are b to a.

APPLIED EXAMPLE 8

Roulette Find the odds in favor of winning a

bet on red in American roulette. What are the odds against winning a bet

on red?

Solution

The probability of winning a bet here—the probability that the ball

lands in a red compartment—is given by P  . Therefore, using Formula (3a),

we see that the odds in favor of winning a bet on red are

or 9 to 10. Next, using (3b), we see that the odds against winning a bet on red are

, or 10 to 9.

Now, suppose the odds in favor of the occurrence of an event are a to b. Then, (3a)

gives

Cross-multiply.

which leads us to the following result.

Formula (4) is often used to determine subjective probabilities, as the next example

shows.

P1E2

a  b

a  1a  b 2P1E2

a  aP1E2 bP1E2

a31  P1E24  bP1E2



P1E2

1  P1E2



38  18

P1E2

1  P1E2



1 

8.2 EXPECTED VALUE 433

Probability of an Event (Given the Odds)

If the odds in favor of an event E occurring are a to b, then the probability of E

occurring is

(4)P1E2

a  b

, event of winning a bet on

red

87533_08_ch8_p417-482 1/30/08 10:09 AM Page 433

EXAMPLE 9

Consider each of the following statements.

a. “The odds that the Dodgers will win the World Series this season are 7 to 5.”

b. “The odds that it will not rain tomorrow are 3 to 2.”

Express each of these odds as a probability of the event occurring.

Solution

a. Using Formula (4) with a  7 and b  5 gives the required probability as

b. Here, the event is that it will not rain tomorrow. Using Formula (4) with a  3

and b  2, we conclude that the probability that it will not rain tomorrow is

3  2



 .6

7  5



⬇ .5833

434 8 PROBABILITY DISTRIBUTIONS AND STATISTICS

PORTFOLIO

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world across a wide array of industries. Statistics play a big

part in helping our clients find solutions to their marketing

problems.

A manufacturer of a brand of juice wanted to know how

its brand was performing overall and how various factors in

the marketplace were affecting consumers’ perceptions and

usage of its brand as well as the juice category overall. This

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introducing new packaging and flavors, and had reduced the

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A year-long tracking study was conducted over the Inter-

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Using statistics, we were able to determine with certainty

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TITLE Senior Project Director

INSTITUTION GfK ARBOR

Explore & Discuss

In the movie Casino, the executive of the Tangiers Casino, Sam Rothstein (Robert DeNiro),

fired the manager of the slot machines in the casino after three gamblers hit three “million

dollar” jackpots in a span of 20 minutes. Rothstein claimed that it was a scam and that

87533_08_ch8_p417-482 1/30/08 10:09 AM Page 434

8.2 EXPECTED VALUE 435

Median and Mode

In addition to the mean, there are two other measures of central tendency of a group

of numerical data: the median and the mode of a group of numbers.

APPLIED EXAMPLE 10

Commuting Times

a. The times, in minutes, Susan took to go to work on nine consecutive

working days were

46 42 49 40 52 48 45 43 50

What is the median of her morning commute times?

b. The times, in minutes, Susan took to return home from work on eight consec-

utive working days were

37 36 39 37 34 38 41 40

What is the median of her evening commute times?

Solution

a. Arranging the numbers in increasing order, we have

40 42 43 45 46 48 49 50 52

Here we have an odd number of entries with the middle number equal to 46,

and this gives the required median.

b. Arranging the numbers in increasing order, we have

34 36 37 37 38 39 40 41

Here the number of entries is even, and the required median is

Mode

The mode of a group of numbers is the number in the group that occurs most

frequently.

37  38

 37.5

Median

The median of a group of numbers arranged in increasing or decreasing order is

(a) the middle number if there is an odd number of entries or (b) the mean of the

two middle numbers if there is an even number of entries.

somebody had gotten into those machines to set the wheels. He was especially annoyed at

the slot machine manager’s assertion that there was no way to determine this. According to

Rothstein the odds of hitting a jackpot in a four-wheel machine is 1 in million, and the

probability of hitting three jackpots in a row is “in the billions.” “It cannot happen! It

will not happen!” To see why Rothstein was so indignant, find the odds of hitting the jack-

pots in three of the machines in quick succession and comment on the likelihood of this

happening.

87533_08_ch8_p417-482 1/30/08 10:09 AM Page 435

1. Find the expected value of a random variable X having the

following probability distribution:

x 4 3 10 1 2

P(X  x) .10 .20 .25 .10 .25 .10

2. The developer of Shoreline Condominiums has provided

the following estimate of the probability that 20, 25, 30, 35,

40, 45, or 50 of the townhouses will be sold within the first

month they are offered for sale.

Units 20 25 30 35 40 45 50

Probability .05 .10 .30 .25 .15 .10 .05

How many townhouses can the developer expect to sell

within the first month they are put on the market?

Solutions to Self-Check Exercises 8.2 can be found on

page 440.

436

8 PROBABILITY DISTRIBUTIONS AND STATISTICS

Note

A group of numerical data may have no mode, a unique mode, or more than one mode.

EXAMPLE 11

Find the mode, if there is one, of the given group of numbers.

a. 1, 2, 3, 4, 6 b. 2, 3, 3, 4, 6, 8 c. 2, 3, 3, 3, 4, 4, 4, 8

Solution

a. The set has no mode because there isn’t a number that occurs more frequently

than the others.

b. The mode is 3 because it occurs more frequently than the others.

c. The modes are 3 and 4 because each number occurs three times.

Of the three measures of central tendency of a group of numerical data, the mean is

by far the most suitable in work that requires mathematical computations.

436 8 PROBABILITY DISTRIBUTIONS AND STATISTICS

1. What is the expected value of a random variable? Give an

example.

2. What is a fair game? Is the game of roulette as played in

American casinos a fair game? Why?

3. a. If the probability of an event E occurring is P(E), what

are the odds in favor of E occurring?

b. If the odds in favor of an event occurring are a to b,

what is the probability of E occurring?

1. During the first year at a university that uses a 4-point

grading system, a freshman took ten 3-credit courses and

received two As, three Bs, four Cs, and one D.

a. Compute this student’s grade-point average.

b. Let the random variable X denote the number of points

corresponding to a given letter grade. Find the probabil-

ity distribution of the random variable X and compute

E(X), the expected value of X.

2. Records kept by the chief dietitian at the university cafete-

ria over a 30-wk period show the following weekly con-

sumption of milk (in gallons).

Milk 200 205 210 215 220

Weeks 34 65 4

Milk 225 230 235 240

Weeks 32 21

a. Find the average number of gallons of milk consumed

per week in the cafeteria.

b. Let the random variable X denote the number of gallons

of milk consumed in a week at the cafeteria. Find the

probability distribution of the random variable X and

compute E(X ), the expected value of X.

8.2 Exercises

8.2 Self-Check Exercises

8.2 Concept Questions

87533_08_ch8_p417-482 1/30/08 10:09 AM Page 436

3. Find the expected value of a random variable X having the

following probability distribution:

x 5 10 1 58

P(X  x) .12 .16 .28 .22 .12 .10

4. Find the expected value of a random variable X having the

following probability distribution:

x 012345

P(X  x)

5. The daily earnings X of an employee who works on a com-

mission basis are given by the following probability distri-

bution. Find the employee’s expected earnings.

x (in $) 0 255075

P(X  x) .07 .12 .17 .14

x (in $) 100 125 150

P(X  x) .28 .18 .04

6. In a four-child family, what is the expected number of

boys? (Assume that the probability of a boy being born is

the same as the probability of a girl being born.)

7. Based on past experience, the manager of the VideoRama

Store has compiled the following table, which gives the

probabilities that a customer who enters the VideoRama

Store will buy 0, 1, 2, 3, or 4 DVDs. How many DVDs can

a customer entering this store be expected to buy?

DVDs 01234

Probability .42 .36 .14 .05 .03

8. If a sample of three batteries is selected from a lot of ten,

of which two are defective, what is the expected number of

defective batteries?

9. A

UTO

CCIDENTS

The number of accidents that occur at a

certain intersection known as “Five Corners” on a Friday

afternoon between the hours of 3 p.m. and 6 p.m., along

with the corresponding probabilities, are shown in the fol-

lowing table. Find the expected number of accidents dur-

ing the period in question.

Accidents 01234

Probability .935 .030 .020 .010 .005

10. E

XPECTED

EMAND

The owner of a newsstand in a college

community estimates the weekly demand for a certain

magazine as follows:

Quantity

Demanded 10 11 12 13 14 15

Probability .05 .15 .25 .30 .20 .05

Find the number of issues of the magazine that the news-

stand owner can expect to sell per week.

11. E

XPECTED

RODUCT

ELIABILITY

A bank has two automatic

tellers at its main office and two at each of its three

branches. The number of machines that break down on a

given day, along with the corresponding probabilities, are

shown in the following table.

Machines That

Break Down 01234

Probability .43 .19 .12 .09 .04

Machines That

Break Down 5678

Probability .03 .03 .02 .05

Find the expected number of machines that will break

down on a given day.

12. E

XPECTED

ALES

The management of the Cambridge Com-

pany has projected the sales of its products (in millions of

dollars) for the upcoming year, with the associated proba-

bilities shown in the following table:

Sales 20 22 24 26 28 30

Probability .05 .10 .35 .30 .15 .05

What does the management expect the sales to be next

year?

13. I

NTEREST

-R

ATE

REDICTION

A panel of 50 economists was

asked to predict the average prime interest rate for the

upcoming year. The results of the survey follow:

Interest Rate, % 4.9 5.0 5.1 5.2 5.3 5.4

Economists 3 8 12 14 8 5

Based on this survey, what does the panel expect the aver-

age prime interest rate to be next year?

14. U

NEMPLOYMENT

ATES

A panel of 64 economists was

asked to predict the average unemployment rate for the

upcoming year. The results of the survey follow:

Unemployment

Rate, % 4.5 4.6 4.7 4.8 4.9 5.0 5.1

Economists 24 82014124

Based on this survey, what does the panel expect the aver-

age unemployment rate to be next year?

15. L

OTTERIES

In a lottery, 5000 tickets are sold for $1 each.

One first prize of $2000, 1 second prize of $500, 3 third

prizes of $100, and 10 consolation prizes of $25 are to be

awarded. What are the expected net earnings of a person

who buys one ticket?

8.2 EXPECTED VALUE 437

87533_08_ch8_p417-482 1/30/08 10:09 AM Page 437

16. L

IFE

NSURANCE

REMIUMS

A man wishes to purchase a

5-yr term-life insurance policy that will pay the beneficiary

$20,000 in the event that the man’s death occurs during the

next 5 yr. Using life insurance tables, he determines that

the probability that he will live another 5 yr is .96. What

is the minimum amount that he can expect to pay for his

premium?

Hint: The minimum premium occurs when the insurance com-

pany’s expected profit is zero.

17. L

IFE

NSURANCE

REMIUMS

A woman purchased a $10,000,

1-yr term-life insurance policy for $130. Assuming that the

probability that she will live another year is .992, find the

company’s expected gain.

18. L

IFE

NSURANCE

OLICIES

As a fringe benefit, Dennis Taylor

receives a $25,000 life insurance policy from his employer.

The probability that Dennis will live another year is .9935.

If he purchases the same coverage for himself, what is

the minimum amount that he can expect to pay for the

policy?

19. E

XPECTED

ROFIT

Max built a spec house at a cost of

$450,000. He estimates that he can sell the house for

$580,000, $570,000, or $560,000, with probabilities .24,

.40, and .36, respectively. What is Max’s expected profit?

20. I

NVESTMENT

NALYSIS

The proprietor of Midland Con-

struction Company has to decide between two projects. He

estimates that the first project will yield a profit of

$180,000 with a probability of .7 or a profit of $150,000

with a probability of .3; the second project will yield a

profit of $220,000 with a probability of .6 or a profit of

$80,000 with a probability of .4. Which project should the

proprietor choose if he wants to maximize his expected

profit?

21. C

ABLE

ELEVISION

The management of MultiVision, a

cable TV company, intends to submit a bid for the cable

television rights in one of two cities, A or B. If the com-

pany obtains the rights to city A, the probability of which

is .2, the estimated profit over the next 10 yr is $10 million;

if the company obtains the rights to city B, the probability

of which is .3, the estimated profit over the next 10 yr is $7

million. The cost of submitting a bid for rights in city A is

$250,000 and that in city B is $200,000. By comparing the

expected profits for each venture, determine whether the

company should bid for the rights in city A or city B.

22. E

XPECTED

UTO

ALES

Roger Hunt intends to purchase one

of two car dealerships currently for sale in a certain city.

Records obtained from each of the two dealers reveal that

their weekly volume of sales, with corresponding probabil-

ities, are as follows:

Dahl Motors

Cars Sold/Week 5678

Probability .05 .09 .14 .24

Cars Sold/Week 9101112

Probability .18 .14 .11 .05

Farthington Auto Sales

Cars Sold/Week 5678910

Probability .08 .21 .31 .24 .10 .06

The average profit/car at Dahl Motors is $362, and the

average profit/car at Farthington Auto Sales is $436.

a. Find the average number of cars sold each week at each

dealership.

b. If Roger’s objective is to purchase the dealership that

generates the higher weekly profit, which dealership

should he purchase? (Compare the expected weekly

profit for each dealership.)

23. E

XPECTED

OME

ALES

Sally Leonard, a real estate broker,

is relocating in a large metropolitan area where she has

received job offers from realty company A and realty com-

pany B. The number of houses she expects to sell in a year

at each firm and the associated probabilities are shown in

the following tables.

Company A

Houses Sold 12 13 14 15 16

Probability .02 .03 .05 .07 .07

Houses Sold 17 18 19 20

Probability .16 .17 .13 .11

Houses Sold 21 22 23 24

Probability .09 .06 .03 .01

Company B

Houses Sold 67 8910

Probability .01 .04 .07 .06 .11

Houses Sold 11 12 13 14

Probability .12 .19 .17 .13

Houses Sold 15 16 17 18

Probability .04 .03 .02 .01

The average price of a house in the locale of company A is

$308,000, whereas the average price of a house in the

locale of company B is $474,000. If Sally will receive a 3%

commission on sales at both companies, which job offer

should she accept to maximize her expected yearly com-

mission?

24. I

NVESTMENT

NALYSIS

Bob, the proprietor of Midway

Lumber, bases his projections for the annual revenues of

the company on the performance of the housing market. He

rates the performance of the market as very strong, strong,

normal, weak, or very weak. For the next year, Bob esti-

mates that the probabilities for these outcomes are .18, .27,

.42, .10, and .03, respectively. He also thinks that the rev-

enues corresponding to these outcomes are $20, $18.8,

$16.2, $14, and $12 million, respectively. What is Bob’s

expected revenue for next year?

25. R

EVENUE

ROJECTION

Maria sees the growth of her busi-

ness for the upcoming year as being tied to the gross

438

8 PROBABILITY DISTRIBUTIONS AND STATISTICS

87533_08_ch8_p417-482 1/30/08 10:09 AM Page 438

domestic product (GDP). She believes that her business

will grow (or contract) at the rate of 5%, 4.5%, 3%, 0%, or

0.5% per year if the GDP grows (or contracts) at the rate

of between 2 and 2.5%, between 1.5 and 2%, between 1

and 1.5%, between 0 and 1%, and between 1 and 0%,

respectively. Maria has decided to assign a probability of

.12, .24, .40, .20, and .04, respectively, to each outcome. At

what rate does Maria expect her business to grow next

year?

26. W

EATHER

REDICTIONS

Suppose the probability that it will

rain tomorrow is .3.

a. What are the odds that it will rain tomorrow?

b. What are the odds that it will not rain tomorrow?

27. R

OULETTE

In American roulette, as described in Example

6, a player may bet on a split (two adjacent numbers). In

this case, if the player bets $1 and either number comes up,

the player wins $17 and gets his $1 back. If neither comes

up, he loses his $1 bet. Find the expected value of the win-

nings on a $1 bet placed on a split.

28. R

OULETTE

If a player placed a $1 bet on red and a $1 bet on

black in a single play in American roulette, what would be

the expected value of his winnings?

29. R

OULETTE

In European roulette, the wheel is divided into

37 compartments numbered 1 through 36 and 0. (In Amer-

ican roulette there are 38 compartments numbered 1

through 36, 0, and 00.) Find the expected value of the win-

nings on a $1 bet placed on red in European roulette.

30. The probability of an event E occurring is .8. What are the

odds in favor of E occurring? What are the odds against E

occurring?

31. The probability of an event E not occurring is .6. What are

the odds in favor of E occurring? What are the odds against

E occurring?

32. The odds in favor of an event E occurring are 9 to 7. What

is the probability of E occurring?

33. The odds against an event E occurring are 2 to 3. What is

the probability of E not occurring?

34. O

DDS

Carmen, a computer sales representative, feels that

the odds are 8 to 5 that she will clinch the sale of a mini-

computer to a certain company. What is the (subjective)

probability that Carmen will make the sale?

35. S

PORTS

Steffi feels that the odds in favor of her winning

her tennis match tomorrow are 7 to 5. What is the (subjec-

tive) probability that she will win her match tomorrow?

36. S

PORTS

If a sports forecaster states that the odds of a cer-

tain boxer winning a match are 4 to 3, what is the (subjec-

tive) probability that the boxer will win the match?

37. O

DDS

Bob, the proprietor of Midland Lumber, feels that

the odds in favor of a business deal going through are 9 to

5. What is the (subjective) probability that this deal will not

materialize?

38. R

OULETTE

a. Show that, for any number c,

E(cX)  cE(X )

b. Use this result to find the expected loss if a gambler bets

$300 on red in a single play in American roulette.

Hint: Use the results of Example 6.

39. E

XAM

CORES

In an examination given to a class of 20 stu-

dents, the following test scores were obtained:

40 45 50 50 55 60 60 75 75 80

80 85 85 85 85 90 90 95 95 100

a. Find the mean (or average) score, the mode, and the me-

dian score.

b. Which of these three measures of central tendency do

you think is the least representative of the set of scores?

40. W

AGE

ATES

The frequency distribution of the hourly

wage rates (in dollars) among blue-collar workers in a cer-

tain factory is given in the following table. Find the mean

(or average) wage rate, the mode, and the median wage rate

of these workers.

Wage Rate 10.70 10.80 10.90 11.00 11.10 11.20

Frequency 60 90 75 120 60 45

41. W

AITING

IMES

Refer to Example 6, Section 8.1. Find the

median of the number of cars waiting in line at the bank’s

drive-in teller at the beginning of each 2-min interval dur-

ing the period in question. Compare you answer to the

mean obtained in Example 1, Section 8.2.

42. S

RANCISCO

EATHER

The normal daily minimum tem-

perature in degrees Fahrenheit for the months of January

through December in San Francisco follows:

46.2 48.4 48.6 49.2 50.7 52.5

53.1 54.2 55.8 54.8 51.5 47.2

Find the average and the median daily minimum tempera-

ture in San Francisco for these months.

Source: San Francisco Convention and Visitors Bureau

43. W

EIGHT OF

OTATO

HIPS

The weights, in ounces, of ten

packages of potato chips are

16.1 16 15.8 16 15.9 16.1 15.9 16 16 16.2

Find the average and the median of these weights.

44. B

OSTON

EATHER

The relative humidity, in percent, in the

morning for the months of January through December in

Boston follows:

68 67 69 69 71 73

74 76 79 77 74 70

Find the average and the median of these humidity read-

ings.

Source: National Weather Service Forecast Office

8.2 EXPECTED VALUE 439

87533_08_ch8_p417-482 1/30/08 10:09 AM Page 439

In Exercises 45 and 46, determine whether the state-

ment is true or false. If it is true, explain why it is true.

If it is false, give an example to show why it is false.

45. A game between two persons is fair if the expected value

to both persons is zero.

46. If the odds in favor of an event E occurring are a to b,

then the probability of E

occurring is b/(a  b).

Variance

The mean, or expected value, of a random variable enables us to express an important

property of the probability distribution associated with the random variable in terms

of a single number. But the knowledge of the location, or central tendency, of a prob-

ability distribution alone is usually not enough to give a reasonably accurate picture

of the probability distribution. Consider, for example, the two probability distributions

whose histograms appear in Figure 9. Both distributions have the same expected

value, or mean, of ␮  4 (the Greek letter ␮ is read “mu”). Note that the probability

distribution with the histogram shown in Figure 9a is closely concentrated about its

mean ␮, whereas the one with the histogram shown in Figure 9b is widely dispersed

or spread about its mean.

As another example, suppose that Olivia has ten packages of brand A potato chips

and ten packages of brand B potato chips. After carefully measuring the weights of

each package, she obtains the following results:

440 8 PROBABILITY DISTRIBUTIONS AND STATISTICS

2. Let X denote the number of townhouses that will be sold

within 1 mo of being put on the market. Then, the number

of townhouses the developer expects to sell within 1 mo is

0.8

 1021.102 11 21.252 1221.102

E1X2 1421.102 13 21.202 1121.252

given by the expected value of X—that is, by

or 34 townhouses.

 34.25

 401.15 2 451.102 501.052

E1X2 201.05 2 251.102 301.302 351.25 2

FIGURE 9

The histograms of two probability

distributions

8.2 Solutions to Self-Check Exercises

8.3 Variance and Standard Deviation

01234567

= 4

␮

01234567

= 4

␮

(a)

(b)

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8.3 VARIANCE AND STANDARD DEVIATION 441

In Example 3, we verify that the mean weights for each of the two brands is

16 ounces. However, a cursory examination of the data now shows that the weights

of the brand B packages exhibit much greater dispersion about the mean than those of

brand A.

One measure of the degree of dispersion, or spread, of a probability distribution

about its mean is given by the variance of the random variable associated with the

probability distribution. A probability distribution with a small spread about its mean

will have a small variance, whereas one with a larger spread will have a larger vari-

ance. Thus, the variance of the random variable associated with the probability distri-

bution whose histogram appears in Figure 9a is smaller than the variance of the ran-

dom variable associated with the probability distribution whose histogram is shown in

Figure 9b (see Example 1). Also, as we will see in Example 3, the variance of the ran-

dom variable associated with the weights of the brand A potato chips is smaller than

that of the random variable associated with the weights of the brand B potato chips.

We now define the variance of a random variable.

Let’s look a little closer at Equation (5). First, note that the numbers

 ␮, x

 ␮, . . . , x

 ␮ (6)

measure the deviations of x

, x

, . . . , x

from ␮, respectively. Thus, the numbers

 ␮)

, (x

 ␮)

, . . . , (x

 ␮)

(7)

measure the squares of the deviations of x

, x

, . . . , x

from ␮, respectively. Next,

by multiplying each of the numbers in (7) by the probability associated with each

value of the random variable X, the numbers are weighted accordingly so that their

sum is a measure of the variance of X about its mean. An attempt to define the vari-

ance of a random variable about its mean in a similar manner using the deviations

in (6), rather than their squares, would not be fruitful since some of the deviations

may be positive whereas others may be negative and hence (because of cancella-

tions) the sum will not give a satisfactory measure of the variance of the random

variable.

Variance of a Random Variable X

Suppose a random variable has the probability distribution

x x

. . .

P(X  x) p

. . .

and expected value

E(X)  ␮

Then the variance of the random variable X is

Var(X)  p

 ␮)

 p

 ␮)



. . .

 p

 ␮)

(5)

Weight in Ounces

Brand A 16.1 16 15.8 16 15.9 16.1 15.9 16 16 16.2

Brand B 16.3 15.7 15.8 16.2 15.9 16.1 15.7 16.2 16 16.1

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