The notion of expectation is not restricted to the random variable itself but can be applied to any function of the random variable. For example, a contractor may be uncertain of the time required to complete a contract. This uncertainty could be represented by a random variable whose possible values are the number of days elapsing from the beginning to the completion of work on the contract. However, the contractor’s primary concern is not with the time taken but rather with the cost of fulfilling the contract. This cost will be a func-tion of the time taken, so in determining expected value of the random variable “cost,” we need to find the expectation of a function of the random variable “time to completion.”
• We expect a larger variance because the probabilities of extreme values 0 and 5 are larger. Note that the mean has increased from 1.95 to 2.15, while the vari-ance has increased from 1.95 to 3.83, reflecting the higher probabilities of more extreme values of X.
Expected Value of Functions of Random Variables
Let X be a discrete random variable with probability distribution P1x2, and let g1X2 be some function of X. Then the expected value, E3g1X24, of that func-tion is defined as follows:
E3g1X24 = a
x
g1x2P1x2 (4.7)
Summary of Properties for Linear Functions of a Random Variable
Let X be a random variable with mean mX and variance s2X, and let a and b be any constant fixed numbers. Define the random variable Y as a + bX. Then, the mean and variance of Y are
mY = E3a + bX4 = a + bmX (4.9)
156 Chapter 4 Discrete Probability Distributions and
s2
Y = Var1a + bX2 = b2s2
X (4.10)
so that the standard deviation of Y is
sY = u b usX
Example 4.5 Total Project Cost (Computations for Functions of Random Variables)
A contractor is interested in the total cost of a project on which she intends to bid. She estimates that materials will cost $25,000 and that her labor will be $900 per day. If the project takes X days to complete, the total labor cost will be 900X dollars, and the total cost of the project (in dollars) will be as follows:
C = 25,000 + 900X
Using her experience the contractor forms probabilities (Table 4.4) of likely comple-tion times for the project.
a. Find the mean and variance for completion time X.
b. Find the mean, variance, and standard deviation for total cost C.
Table 4.4 Probability Distribution for Completion Times
COMPLETION TIME x (DAYS) 10 11 12 13 14
Probability 0.1 0.3 0.3 0.2 0.1
Solution
a. The mean and variance for completion time X can be found using Equations 4.4 and 4.5.
mX = E3X4 = a
x
xP1x2
= 110210.12 + 111210.32 + 112210.32 + 113210.22 + 114210.12 = 11.9 days And
s2x = E31X - mx224 = a
x 1x - mx22P1x2
= 110 - 11.92210.12 + 111 - 11.92210.32 + g + 114 - 11.92210.12 = 1.29 b. The mean, variance, and standard deviation of total cost, C, are obtained using
Equations 4.9 and 4.10.
The mean is as follows:
mC = E325,000 + 900X4 = 125,000 + 900mX2 = 25,000 + 19002111.92 = +35,710 The variance is as follows:
s2
C = Var125,000 + 900X2 = 190022s2
X = 1810,000211.292 = 1,044,900 The standard deviation is as follows:
sC = 2s2C = +1,022.20
Exercises 157 Three special examples of the linear function W = a + bX are important. The first ex-ample considers a constant function, W = a, for any constant a. In this situation the coef-ficient b = 0. In the second example a = 0, giving W = bX. The expected value and the variance for these functions are defined by Equations 4.11 and 4.12. The third example is sig-nificant in later chapters. The mean and variance of this special linear function are defined by Equations 4.13 and 4.14. Thus, subtracting its mean from a random variable and dividing by its standard deviation yields a random variable with mean 0 and standard deviation 1.
Summary Results for the Mean and Variance of Special Linear Functions
a. Let b = 0 in the linear function W = a + bX. Then let W = a (for any con-stant a).
E3a4 = a and Var1a2 = 0 (4.11)
If a random variable always takes the value a, it will have a mean a and a variance 0.
b. Let a = 0 in the linear function W = a + bX. Then let W = bX.
E3bX4 = bmX and Var1bX2 = b2s2
X (4.12)
c. To find the mean and variance of
Z = X- mX
sX
let a = -mX>sX and b = 1>sX in the linear function Z = a + bX. Then Z = a + bX = X - mX
sX = X sX - mX
sX so that
EcX - mX
sX d = mX sX - 1
sXmX = 0 (4.13) and
VaraX - mX
sX b = 1 s2
X
s2X = 1 (4.14)
E
XERCISESBasic Exercises
4.15 Consider the probability distribution function.
x 0 1
Probability 0.40 0.60
a. Graph the probability distribution function.
b. Calculate and graph the cumulative probability distribution.
c. Find the mean of the random variable X.
d. Find the variance of X.
4.16 Given the probability distribution function:
x 0 1 2
Probability 0.25 0.50 0.25
a. Graph the probability distribution function.
b. Calculate and graph the cumulative probability distribution.
c. Find the mean of the random variable X.
d. Find the variance of X.
4.17 Consider the probability distribution function
x 0 1
Probability 0.50 0.50
a. Graph the probability distribution function.
b. Calculate and graph the cumulative probability distribution.
c. Find the mean of the random variable X.
d. Find the variance of X.
158 Chapter 4 Discrete Probability Distributions
4.18 An automobile dealer calculates the proportion of new cars sold that have been returned a various num-bers of times for the correction of defects during the warranty period. The results are shown in the follow-ing table.
Number of returns 0 1 2 3 4
Proportion 0.28 0.36 0.23 0.09 0.04
a. Graph the probability distribution function.
b. Calculate and graph the cumulative probability distribution.
c. Find the mean of the number of returns of an automobile for corrections for defects during the warranty period.
d. Find the variance of the number of returns of an au-tomobile for corrections for defects during the war-ranty period.
4.19 A company specializes in installing and servicing central-heating furnaces. In the prewinter period, ser-vice calls may result in an order for a new furnace. The following table shows estimated probabilities for the numbers of new furnace orders generated in this way in the last two weeks of September.
Number of orders 0 1 2 3 4 5
Probability 0.10 0.14 0.26 0.28 0.15 0.07 a. Graph the probability distribution function.
b. Calculate and graph the cumulative probability distribution.
c. Find the probability that at least 3 orders will be generated in this period.
d. Find the mean of the number of orders for new furnaces in this 2-week period.
e. Find the standard deviation of the number of orders for new furnaces in this 2-week period.
Application Exercises
4.20 Forest Green Brown, Inc., produces bags of cypress mulch. The weight in pounds per bag varies, as indi-cated in the accompanying table.
Weight in pounds 44 45 46 47 48 49 50 Proportion of bags 0.04 0.13 0.21 0.29 0.20 0.10 0.03
a. Graph the probability distribution.
b. Calculate and graph the cumulative probability distribution.
c. What is the probability that a randomly chosen bag will contain more than 45 and less than 49 pounds of mulch (inclusive)?
d. Two packages are chosen at random. What is the probability that at least one of them contains at least 47 pounds?
e. Compute—using a computer—the mean and stan-dard deviation of the weight per bag.
f. The cost (in cents) of producing a bag of mulch is 75+ 2X, where X is the number of pounds per bag. The revenue from selling the bag, regardless of
weight, is $2.50. If profit is defined as the difference between revenue and cost, find the mean and stan-dard deviation of profit per bag.
4.21 A municipal bus company has started operations in a new subdivision. Records were kept on the numbers of riders on one bus route during the early-morning weekday service. The accompanying table shows pro-portions over all weekdays.
Number of riders 20 21 22 23 24 25 26 27 Proportion 0.02 0.12 0.23 0.31 0.19 0.08 0.03 0.02
a. Graph the probability distribution.
b. Calculate and graph the cumulative probability distribution.
c. What is the probability that on a randomly chosen weekday there will be at least 24 riders from the subdivision on this service?
d. Two weekdays are chosen at random. What is the probability that on both of these days there will be fewer than 23 riders from the subdivision on this service?
e. Find the mean and standard deviation of the num-ber of riders from this subdivision on this service on a weekday.
f. If the cost of a ride is $1.50, find the mean and stan-dard deviation of the total payments of riders from this subdivision on this service on a weekday.
4.22 a. A very large shipment of parts contains 10% de-fectives. Two parts are chosen at random from the shipment and checked. Let the random vari-able X denote the number of defectives found.
Find the probability distribution of this random variable.
b. A shipment of 20 parts contains 2 defectives. Two parts are chosen at random from the shipment and checked. Let the random variable Y denote the number of defectives found. Find the probability distribution of this random variable. Explain why your answer is different from that for part (a).
c. Find the mean and variance of the random variable X in part (a).
d. Find the mean and variance of the random variable Y in part (b).
4.23 A student needs to know details of a class assign-ment that is due the next day and decides to call fel-low class members for this information. She believes that for any particular call, the probability of obtain-ing the necessary information is 0.40. She decides to continue calling class members until the infor-mation is obtained. But her cell phone battery will not allow more than 8 calls. Let the random variable X denote the number of calls needed to obtain the information.
a. Find the probability distribution of X.
b. Find the cumulative probability distribution of X.
c. Find the probability that at least three calls are required.
4.4 Binomial Distribution 159
4.4 B
INOMIALD
ISTRIBUTIONWe now develop the binomial probability distribution, which is used extensively in many applied business and economic problems. Our approach begins with the Bernoulli model, which is a building block for the binomial. Consider a random experiment that can give rise to just two possible mutually exclusive and collectively exhaustive outcomes, which for convenience we label “success” and “failure.” Let P denote the probability of success, and, the probability of failure 11 - P2. Then, define the random variable X so that X takes the value 1 if the outcome of the experiment is success and 0 otherwise. The prob-ability distribution of this random variable is then
P102 = 11 - P2 and P112 = P
This distribution is known as the Bernoulli distribution. Its mean and variance can be found by direct application of the equations in Section 4.3.
4.24 Your school Ping-Pong team is not performing very well this season. After some rough calculations, you found out that your team’s probability of winning a game is about 0.45. A fellow team member wants to know more and asked you also to determine the following.
a. The probability of the team winning 2 games out of 5.
b. The probability of winning 10 times out of 25.
4.25 A professor teaches a large class and has scheduled an examination for 7:00 p.m. in a different classroom. She estimates the probabilities in the table for the number of students who will call her at home in the hour before the examination asking where the exam will be held.
Number of calls 0 1 2 3 4 5
Probability 0.10 0.15 0.19 0.26 0.19 0.11 Find the mean and standard deviation of the number of calls.
4.26 Students in a large accounting class were asked to rate the course by assigning a score of 1, 2, 3, 4, or 5 to the course. A higher score indicates that the students re-ceived greater value from the course. The accompa-nying table shows proportions of students rating the course in each category.
Rating 1 2 3 4 5
Proportion 0.07 0.19 0.28 0.30 0.16 Find the mean and standard deviation of the ratings.
4.27 A store owner stocks an out-of-town newspaper that is sometimes requested by a small number of custom-ers. Each copy of this newspaper costs her 70 cents, and she sells them for 90 cents each. Any copies left over at the end of the day have no value and are de-stroyed. Any requests for copies that cannot be met because stocks have been exhausted are considered by the store owner as a loss of 5 cents in goodwill. The probability distribution of the number of requests for
the newspaper in a day is shown in the accompany-ing table. If the store owner defines total daily profit as total revenue from newspaper sales, less total cost of newspapers ordered, less goodwill loss from unsatis-fied demand, what is the expected profit if four news-papers are order?
Number of requests 0 1 2 3 4 5 Probability 0.12 0.16 0.18 0.32 0.14 0.08
4.28 A factory manager is considering whether to replace a temperamental machine. A review of past records indicates the following probability distribution for the number of breakdowns of this machine in a week.
Number of breakdowns 0 1 2 3 4
Probability 0.10 0.26 0.42 0.16 0.06
a. Find the mean and standard deviation of the num-ber of weekly breakdowns.
b. It is estimated that each breakdown costs the com-pany $1,500 in lost output. Find the mean and stan-dard deviation of the weekly cost to the company from breakdowns of this machine.
4.29 An investor is considering three strategies for a $1,000 investment. The probable returns are estimated as follows:
• Strategy 1: A profit of $10,000 with probability 0.15 and a loss of $1,000 with probability 0.85
• Strategy 2: A profit of $1,000 with probability 0.50, a profit of $500 with probability 0.30, and a loss of
$500 with probability 0.20
• Strategy 3: A certain profit of $400
Which strategy has the highest expected profit? Ex-plain why you would or would not advise the investor to adopt this strategy.
160 Chapter 4 Discrete Probability Distributions