Download Chapter 01 Discrete Probability Distribution Exercises

Chapter 01
Discrete Probability Distribution
Exercises
Q.1:
Consider the experiment of tossing a coin twice.
a) List the experimental outcomes.
b) Define a random variable that represents the number of heads occurring on the two tosses.
c) Show what value the random variable would assume for each of the experimental
outcomes.
d. Is this random variable discrete or continuous?
Q.2
Three students scheduled interviews for summer employment at the Brookwood Institute. In
each case the interview results in either an offer for a position or no offer. Experimental
outcomes are defined in terms of the results of the three interviews.
a) List the experimental outcomes.
b) Define a random variable that represents the number of offers made. Is the random
variable continuous?
c)Show the value of the random variable for each of the experimental outcomes.
Q.3.
Listed is a series of experiments and associated random variables. In each case, identify the values
that the random variable can assume and state whether the random variable is discrete or
continuous.
Experiment
a. Take a 20-question examination
Random Variable (x)
Number of questions answered correctly
b. Observe cars arriving at a tollbooth for one
hour
Number of cars arriving at tollbooth
c. Audit 50 tax returns
Number of returns containing errors
d. Observe an employee’s work
Number of nonproductive hours in an eight
hour workday
e. Weigh a shipment of goods
Number of pounds
Commercial Statistics
Q.4.
The probability distribution for the random variable x follows.
X
20
25
30
35
f(x)
.20
.15
.25
.40
a. Is this probability distribution valid? Explain.
b. What is the probability that x = 30?
c. What is the probability that x is less than or equal to 25?
d. What is the probability that x is greater than 30?
Q.5.
The following data were collected by counting the number of operating rooms in use at
Tampa General Hospital over a 20-day period: On 3 of the days only one operating room
was used, on 5 of the days two were used, on 8 of the days three were used, and on 4 days
all four of the hospital’s operating rooms were used.
a. Use the relative frequency approach to construct a probability distribution for the
number of operating rooms in use on any given day.
b. Draw a graph of the probability distribution.
c. Show that your probability distribution satisfies the required conditions for a valid discrete
probability distribution.
Q.6:
The percent frequency distributions of job satisfaction scores for a sample of information
systems (IS) senior executives and middle managers are as follows. The scores range from a
low of 1 (very dissatisfied) to a high of 5 (very satisfied).
Job Satisfaction
Score
1
2
3
4
5
IS Senior
Executives (%)
5
9
3
42
41
IS Middle
Managers (%)
4
10
12
46
28
a. Develop a probability distribution for the job satisfaction score of a senior executive.
b. Develop a probability distribution for the job satisfaction score of a middle manager.
c. What is the probability that a senior executive will report a job satisfaction score of 4 or 5?
d. What is the probability that a middle manager is very satisfied?
Commercial Statistics
Q.7.
For unemployed persons in the United States, the average number of months of
unemployment at the end of December 2009 was approximately seven months (Bureau of
Labor Statistics, January 2010). Suppose the following data are for a particular region in
upstate New York. The values in the first column show the number of months unemployed
and the values in the second column show the corresponding number of unemployed persons.
Months
Unemployed
1
2
3
4
5
6
7
8
9
10
Number
Unemployed
1029
1686
2269
2675
3487
4652
4145
3587
2325
1120
Let x be a random variable indicating the number of months a person is unemployed.
a. Use the data to develop a probability distribution for x.
b. Show that your probability distribution satisfies the conditions for a valid discrete
probability distribution.
c. What is the probability that a person is unemployed for two months or less? Unemployed
for more than two months?
d. What is the probability that a person is unemployed for more than six months?
Q.8:
A technician services mailing machines at companies in the Phoenix area. Depending on
the type of malfunction, the service call can take one, two, three, or four hours. The different
types of malfunctions occur at about the same frequency.
a. Develop a probability distribution for the duration of a service call.
b. Draw a graph of the probability distribution.
c. Show that your probability distribution satisfies the conditions required for a discrete
probability function.
d. What is the probability that a service call will take three hours?
Q.9:
Apsychologist determined that the number of sessions required to obtain the trust of a new
patient is either 1, 2, or 3. Let x be a random variable indicating the number of sessions
required to gain the patient’s trust. The following probability function has been proposed.
a. Is this probability function valid? Explain.
b. What is the probability that it takes exactly two sessions to gain the patient’s trust?
c. What is the probability that it takes at least two sessions to gain the patient’s trust?
Commercial Statistics
Q.10:
The following table provides a probability distribution for the random variable x.
X
3
6
9
f(x)
.25
.50
.25
a. Compute E(x), the expected value of x.
b. Compute σ2, the variance of x.
c. Compute σ, the standard deviation of x.
Q.11:
The following table provides a probability distribution for the random variable y.
Y
2
4
7
8
f( y)
.20
.30
.40
.10
a) Compute E( y).
b) Compute Var( y) and σ
Q.12:
The following probability distributions of job satisfaction scores for a sample of information
systems (IS) senior executives and middle managers range from a low of 1 (very dissatisfied)
to a high of 5 (very satisfied).
Probability
Job Satisfaction
IS Senior
IS Middle
Score
Executives
Managers
1
.05
.04
2
.09
.10
3
.03
.12
4
.42
.46
5
.41
.28
a. What is the expected value of the job satisfaction score for senior executives?
b. What is the expected value of the job satisfaction score for middle managers?
c. Compute the variance of job satisfaction scores for executives and middle managers.
d. Compute the standard deviation of job satisfaction scores for both probability distributions.
Q.13:
Consider a binomial experiment with two trials and p = .4.
a. Draw a tree diagram for this experiment (see Figure 5.3).
b. Compute the probability of one success, f (1).
c. Compute f (0).
d. Compute f (2).
e. Compute the probability of at least one success.
f. Compute the expected value, variance, and standard deviation.
Commercial Statistics
Q.14:
In San Francisco, 30% of workers take public transportation daily.
a. In a sample of 10 workers, what is the probability that exactly 3 workers take public
transportation daily?
b. In a sample of 10 workers, what is the probability that at least 3 workers take public
transportation daily?
Q.15:
When a new machine is functioning properly, only 3% of the items produced are defective.
Assume that we will randomly select two parts produced on the machine and that we are
interested in the number of defective parts found.
a) Describe the conditions under which this situation would be a binomial experiment.
b) Draw a tree diagram similar to Figure 5.3 showing this problem as a two-trial experiment.
c) How many experimental outcomes result in exactly one defect being found?
d) Compute the probabilities associated with finding no defects, exactly one defect, and
two defects.
Q.16:
A university found that 20% of its students withdraw without completing the introductory
statistics course. Assume that 20 students registered for the course.
a. Compute the probability that two or fewer will withdraw.
b. Compute the probability that exactly four will withdraw.
c. Compute the probability that more than three will withdraw.
d. Compute the expected number of withdrawals.
Q.17:
Consider a Poisson distribution with μ _ 3.
a. Write the appropriate Poisson probability function.
b. Compute f (2).
c. Compute f (1).
d. Compute P(x _ 2).
Q.18:
Consider a Poisson distribution with a mean of two occurrences per time period.
a) Write the appropriate Poisson probability function.
b) What is the expected number of occurrences in three time periods?
c) Write the appropriate Poisson probability function to determine the probability of x
occurrences in three time periods.
d) Compute the probability of two occurrences in one time period.
e) Compute the probability of six occurrences in three time periods.
f) Compute the probability of five occurrences in two time periods
Commercial Statistics
Q.19:
More than 50 million guests stay at bed and breakfasts (B&Bs) each year. The website for
the Bed and Breakfast Inns of North America, which averages seven visitors per minute,
enables many B&Bs to attract guests (Time, September 2001).
a) Compute the probability of no website visitors in a one-minute period.
b) Compute the probability of two or more website visitors in a one-minute period.
c) Compute the probability of one or more website visitors in a 30-second period.
d) Compute the probability of five or more website visitors in a one-minute period.
Q.20
Airline passengers arrive randomly and independently at the passenger-screening facility
at a major international airport. The mean arrival rate is 10 passengers per minute.
a) Compute the probability of no arrivals in a one-minute period.
b) Compute the probability that three or fewer passengers arrive in a one-minute period.
c) Compute the probability of no arrivals in a 15-second period.
d) Compute the probability of at least one arrival in a 15-second period
Q.21
Suppose N = 10 and r = 3. Compute the hypergeometric probabilities for the following
values of n and x.
a. n = 4, x =1
b. n = 2, x = 2
c. n = 2, x = 0
Q.22
Axline Computers manufactures personal computers at two plants, one in Texas and the
other in Hawaii. The Texas plant has 40 employees; the Hawaii plant has 20. A random
sample of 10 employees is to be asked to fill out a benefits questionnaire.
a) What is the probability that none of the employees in the sample work at the plant in
Hawaii?
b) What is the probability that one of the employees in the sample works at the plant in
Hawaii?
c) What is the probability that two or more of the employees in the sample work at the
plant in Hawaii?
d. What is the probability that nine of the employees in the sample work at the plant in
Texas?
Commercial Statistics