Deck 14: Markov Analysis

Question 1

For any absorbing state, the probability that a state will remain unchanged in the future is one.

Answer 1

True

Question 2

In Markov analysis, the row elements of the transition matrix must sum to 1.

Answer 2

True

Question 3

Once a Markov process is in equilibrium, it stays in equilibrium.

Answer 3

True

Question 4

Markov analysis assumes that while a member of one state may move to a different state over time, the overall makeup of the system will remain the same.

Question 5

In Markov analysis it is assumed that states are both mutually exclusive and collectively exhaustive.

Question 6

(n + 1)= nP

Question 7

Markov analysis assumes that there are a limited number of states in the system.

Question 8

Equilibrium state probabilities may be estimated by using Markov analysis for a large number of periods.

Question 9

In Markov analysis, initial-state probability values determine equilibrium conditions.

Question 10

In Markov analysis, the transition probability Pij represents the conditional probability of being in state i in the future given the current state of j.

Question 11

Creating the fundamental matrix requires a partition of the matrix of transition.

Question 12

An equilibrium condition exists if the state probabilities for a future period are the same as the state probabilities for a previous period.

Question 13

The vector of state probabilities gives the probability of being in particular states at a particular point in time.

Question 14

Markov analysis is a technique that deals with the probabilities of future occurrences by analyzing currently known probabilities.

Question 15

The probabilities in any column of the matrix of transition probabilities will always sum to one.

Question 16

The vector of state probabilities for any period is equal to the vector of state probabilities for the preceding period multiplied by the matrix of transition probabilities.

Question 17

When absorbing states exist, the fundamental matrix is used to compute equilibrium conditions.

Question 18

The four basic assumptions of Markov analysis are:
There are a limited or finite number of possible states.
The probability of changing states remains the same over time.
A future state is predictable from previous state and transition matrix.
The size and makeup of the system are constant during analysis.

Question 19

"Events" are used to identify all possible conditions of a process or a system.

Question 20

In the matrix of transition probabilities, P_ij is the conditional probability of being in state i in the future, given the current state j.

Question 21

In Markov analysis, the vector of state probabilities represents the market shares.

Question 22

One of the purposes of Markov analysis is to predict the future.

Question 23

One of the problems with using the Markov model to study population shifts is that we must assume that the reasons for moving from one state to another remain the same over time.

Question 24

A matrix of transition probabilities is of dimension P_mn.The vector of current market share must be of dimension π_m1.

Question 25

The matrix of transition probabilities gives the conditional probabilities of moving from one state to another.

Question 26

Occasionally, a state is entered that will not allow going to any other state in the future.This is called

A)stability dependency.
B)market saturation.
C)incidental mobility.
D)an absorbing state.

Question 27

A collection of all state probabilities for a given system at any given period of time is called the

A)transition probabilities.
B)vector of state probabilities.
C)fundamental matrix.
D)equilibrium condition.

Question 28

If we are in any period n, we can compute the state probabilities for period n + 1 as π(n + 1)= π(n)P

Question 29

In Markov analysis, we also assume that the states are

A)collectively exhaustive.
B)mutually exclusive.
C)independent.
D)Both collectively exhaustive and mutually exclusive.

Question 30

Markov analysis is a technique that deals with the probabilities of future occurrences by

A)using the simplex solution method.
B)analyzing currently known probabilities.
C)statistical sampling.
D)the minimal spanning tree.

Question 31

A Markov process could be used as a model of how a disease progresses from one set of symptoms to another.

Question 32

In Markov analysis, the likelihood that any system will change from one period to the next is revealed by the

A)cross-elasticities.
B)fundamental matrix.
C)matrix of transition probabilities.
D)vector of state probabilities.

Question 33

In a matrix of transition probabilities

A)the probabilities for any column will sum to 1.
B)the probabilities for any row will sum to 1.
C)the probabilities for any column are mutually exclusive and collectively exhaustive.
D)the probabilities of all columns will sum to 1.

Question 34

A state probability when equilibrium has been reached is called

A)state probability.
B)prior probability.
C)steady state probability.
D)joint probability.

Question 35

In a tree diagram, the numbers associated with the arcs moving from one state to the next would be represented in the matrix of transition probabilities.

Question 36

The probability that we will be in a future state, given a current or existing state, is called

A)state probability.
B)prior probability.
C)steady state probability.
D)transition probability.

Question 37

The state probabilities for n periods in the future can be obtained from the current state probabilities and the matrix of equilibrium probabilities.

Question 38

Collectively exhaustive means that a system can be in only one state at any point in time.

Question 39

If you are in an absorbing state, you cannot go to another state in the future.

Question 40

Which of the following is not an assumption of Markov processes?

A)The state variable is discrete.
B)There are a limited number of possible states.
C)The probability of changing states remains the same over time.
D)We can predict any future state from the previous state and the matrix of transition probabilities.

Question 41

The copy machine in an office is very unreliable.If it was working yesterday, there is an 80% chance it will work today.If it was not working yesterday, there is a 10% chance it will work today.If it is not working today, what is the probability that it will be working 2 days from now?

A)0.16
B)0.17
C)0.34
D)0.66

Question 42

The initial values for the state probabilities

A)are always greater than the equilibrium state probabilities.
B)are always less than the equilibrium state probabilities.
C)do not influence the equilibrium state probabilities.
D)heavily influence the equilibrium state probabilities.

Question 43

What do we do when solving for equilibrium conditions?

A)conduct a statistical t-test
B)drop one equation
C)partition the matrix of transition probabilities
D)subtract matrix B from the identity matrix

Question 44

If we want to use Markov analysis to study market shares for competitive businesses

A)it is an inappropriate study.
B)simply replace the probabilities with market shares.
C)it can only accommodate one new business each period.
D)only constant changes in the matrix of transition probabilities can be handled in the simple model.

Question 45

The weather is becoming important to you since you would like to go on a picnic today.If it was sunny yesterday, there is a 70% chance it will be sunny today.If it was raining yesterday, there is a 30% chance it will be sunny today.What is the probability it will be rainy today, if it was sunny yesterday?

A)0.1
B)0.2
C)0.3
D)0.4

Question 46

The copy machine in an office is very unreliable.If it was working yesterday, there is an 80% chance it will work today.If it was not working yesterday, there is a 10% chance it will work today.If it is working today, what is the probability that it will be working 2 days from now?

A)0.16
B)0.64
C)0.66
D)0.80

Question 47

Table 14-1
The following data consists of a matrix of transition probabilities (P)of three competing companies, and the initial market share π(0).Assume that each state represents a company (Company 1, Company 2, Company 3, respectively)and the transition probabilities represent changes from one month to the next.
P = π(0)= (0.3, 0.6, 0.1)
Using the data in Table 14-1, and assuming that the transition probabilities do not change, in the long run what market share would Company 2 expect to reach? (Rounded to two decimal places.)

A)0.30
B)0.32
C)0.39
D)0.60

Question 48

In Markov analysis, to find the vector of state probabilities for any period

A)one should find them empirically.
B)subtract the product of the numbers on the primary diagonal from the product of the numbers on the secondary diagonal.
C)find the product of the vector of state probabilities for the preceding period and the matrix of transition probabilities.
D)find the product of the vectors of state probabilities for the two preceding periods.

Question 49

In a(n)________ state, you cannot go to another state in the future.

A)transition
B)equilibrium
C)locked
D)absorbing

Question 50

The matrix that is needed to compute equilibrium conditions when absorbing states are involved is called a(n)

A)transition matrix.
B)fundamental matrix.
C)equilibrium matrix.
D)absorbing matrix.

Question 51

If in an absorbing state, the probability of being in an absorbing state in the future is

A)25%.
B)50%.
C)75%.
D)100%.

Question 52

To find the equilibrium state in Markov analysis

A)it is necessary to know both the vector of state probabilities and the matrix of transition probabilities.
B)it is necessary only to know the matrix of transition probabilities.
C)it is necessary only to know the vector of state probabilities for the initial period.
D)one should develop a table of state probabilities over time and then determine the equilibrium conditions empirically.

Question 53

At equilibrium

A)state probabilities for the next period equal the state probabilities for this period.
B)the state probabilities are all equal to each other.
C)the matrix of transition probabilities is symmetrical.
D)the vector of state probabilities is symmetrical.

Question 54

Where P is the matrix of transition probabilities, π(4)=

A)π(3)P P P.
B)π(3)P P.
C)π(2)P P P.
D)π(1)P P P.

Question 55

Table 14-1
The following data consists of a matrix of transition probabilities (P)of three competing companies, and the initial market share π(0).Assume that each state represents a company (Company 1, Company 2, Company 3, respectively)and the transition probabilities represent changes from one month to the next.
P = π(0)= (0.3, 0.6, 0.1)
Using the data in Table 14-1, determine Company 2's estimated market share in the next period.

A)0.26
B)0.27
C)0.28
D)0.29

Question 56

The weather is becoming important to you since you would like to go on a picnic today.If it was sunny yesterday, there is a 70% chance it will be sunny today.If it was raining yesterday, there is a 30% chance it will be sunny today.If the probability that it was raining yesterday is 0.25, what is the probability that it will rain today?

A)0.1
B)0.3
C)0.4
D)0.7

Question 57

Table 14-1
The following data consists of a matrix of transition probabilities (P)of three competing companies, and the initial market share π(0).Assume that each state represents a company (Company 1, Company 2, Company 3, respectively)and the transition probabilities represent changes from one month to the next.
P = π(0)= (0.3, 0.6, 0.1)
Using the data in Table 14-1, determine Company 1's estimated market share in the next period.

A)0.10
B)0.20
C)0.42
D)0.47

Question 58

The weather is becoming important to you since you would like to go on a picnic today.If it was sunny yesterday, there is a 65% chance it will be sunny today.If it was raining yesterday, there is a 30% chance it will be sunny today.If the probability that it was raining yesterday is 0.4, what is the probability that it will be sunny today?

A)0.650
B)0.390
C)0.510
D)0.490

Question 59

A state that when entered, cannot be left is called

A)transient.
B)recurrent.
C)absorbing.
D)steady.

Question 60

In the long run, in Markov analysis

A)all state probabilities will eventually become zeros or ones.
B)generally, the vector of state probabilities, when multiplied by the matrix of transition probabilities, will yield the same vector of state probabilities.
C)the matrix of transition probabilities will change to an equilibrium state.
D)the matrices will become inverted.

Question 61

Table 14-4
Cuthbert Wylinghauser is a scheduler of transportation for the state of Delirium.This state contains three cities: Chaos (C1), Frenzy (C2), and Tremor (C3).A transition matrix, indicating the probability that a resident in one city will travel to another, is given below.Cuthbert's job is to schedule the required number of seats, one to each person making the trip (transition), on a daily basis.
C F T
Transition matrix:

π(0)= [100, 100, 100]
Using the data given in Table 14-4, find the equilibrium travel population for Frenzy (rounded to the nearest whole person).

A)126
B)95
C)79
D)100

Question 62

Table 14-4
Cuthbert Wylinghauser is a scheduler of transportation for the state of Delirium.This state contains three cities: Chaos (C1), Frenzy (C2), and Tremor (C3).A transition matrix, indicating the probability that a resident in one city will travel to another, is given below.Cuthbert's job is to schedule the required number of seats, one to each person making the trip (transition), on a daily basis.
C F T
Transition matrix:

π(0)= [100, 100, 100]
Using the data given in Table 14-4, how many people can we expect to find in each city tomorrow evening?

A)Chaos = 90, Frenzy = 110, Tremor = 100
B)Chaos = 110, Frenzy = 100, Tremor = 90
C)Chaos = 80, Frenzy = 90, Tremor = 130
D)Chaos = 100, Frenzy = 130, Tremor = 70

Question 63

Table 14-5
The following data consists of a matrix of transition probabilities (P)of three potential diseases, and the initial incidence of each disease π(0).Assume that each state represents a disease (Disease 1, Disease 2, Disease 3, respectively)and the transition probabilities represent changes from one checkup to the next.
P =

π(0)= (.3, .3, .4)
Using the data in Table 14-5, determine Disease 1's estimated incidence in the next period.

A)0.32
B)0.34
C)0.36
D)0.38

Question 64

Table 14-6
The following data consists of a matrix of transition probabilities (P)of four majors in the College of Business, and the initial proportion of students in each major π(0).Assume that each state represents a major and the transition probabilities represent changes from one major to the next after taking the introductory class in each discipline.
P =

π(0)= (.4, .3, .2, .1)
Using the data in Table 14-6, which major will end up with the greatest proportion of students?

A)Major 1
B)Major 2
C)Major 3
D)Major 4

Question 65

Table 14-3
The following data consists of a matrix of transition probabilities (P)of three office locations (A,B,C)within a large company and how employees shift from one location to the other from year to year.The company CEO would like to understand the movement of employees over time and the long-run proportion of employees in each location.Assume that there is always a total of 3000 employees.
A B C
P = π(0)= [1000, 1000, 1000)
Using the data given in Table 14-3, how many employees do we expect in location B one year from now?

A)1000
B)1400
C)1500
D)800

Question 66

Table 14-2
The following data consists of a matrix of transition probabilities (P)of three competing retailers, the initial market share π(0).Assume that each state represents a retailer (Retailer 1, Retailer 2, Retailer 3, respectively)and the transition probabilities represent changes from one month to the next.
P = π(0)= (0.3, 0.6, 0.1)
Using the data given in Table 14-2, what is the equilibrium market share?

A)(0.55, 0.33, 0.12)
B)(0.44, 0.43, 0.12
C)(0.55, 0.12, 0.33)
D)(0.47, 0.40, 0.13)

Question 67

Table 14-5
The following data consists of a matrix of transition probabilities (P)of three potential diseases, and the initial incidence of each disease π(0).Assume that each state represents a disease (Disease 1, Disease 2, Disease 3, respectively)and the transition probabilities represent changes from one checkup to the next.
P =

π(0)= (.3, .3, .4)
Using the data in Table 14-5, which disease will have the highest incidence when absorbing states are reached?

A)Disease 1
B)Disease 2
C)Disease 3
D)Disease 1 and Disease 3 are tied for the highest incidence

Question 68

Table 14-3
The following data consists of a matrix of transition probabilities (P)of three office locations (A,B,C)within a large company and how employees shift from one location to the other from year to year.The company CEO would like to understand the movement of employees over time and the long-run proportion of employees in each location.Assume that there is always a total of 3000 employees.
A B C
P = π(0)= [1000, 1000, 1000)
Using the data given in Table 14-3, how many employees do we expect in location B two years from now?

A)1420
B)1400
C)10090
D)820

Question 69

Table 14-3
The following data consists of a matrix of transition probabilities (P)of three office locations (A,B,C)within a large company and how employees shift from one location to the other from year to year.The company CEO would like to understand the movement of employees over time and the long-run proportion of employees in each location.Assume that there is always a total of 3000 employees.
A B C
P = π(0)= [1000, 1000, 1000)
Using the data given in Table 14-3, how many employees do we expect in location A one year from now?

A)1000
B)1400
C)1500
D)800

Question 70

Table 14-6
The following data consists of a matrix of transition probabilities (P)of four majors in the College of Business, and the initial proportion of students in each major π(0).Assume that each state represents a major and the transition probabilities represent changes from one major to the next after taking the introductory class in each discipline.
P =

π(0)= (.4, .3, .2, .1)
Using the data in Table 14-6, determine Major 1's estimated popularity after students have taken the first introductory course.

A)0.425
B)0.385
C)0.365
D)0.415

Question 71

Table 14-2
The following data consists of a matrix of transition probabilities (P)of three competing retailers, the initial market share π(0).Assume that each state represents a retailer (Retailer 1, Retailer 2, Retailer 3, respectively)and the transition probabilities represent changes from one month to the next.
P = π(0)= (0.3, 0.6, 0.1)
Using the data given in Table 14-2, find the market shares for the three retailers in month 1.

A)π(1)= (0.09, 0.42, 0.49)
B)π(1)= (0.55, 0.33, 0.12)
C)π(1)= (0.18, 0.12, 0.70)
D)π(1)= (0.55, 0.12, 0.33)

Question 72

Table 14-3
The following data consists of a matrix of transition probabilities (P)of three office locations (A,B,C)within a large company and how employees shift from one location to the other from year to year.The company CEO would like to understand the movement of employees over time and the long-run proportion of employees in each location.Assume that there is always a total of 3000 employees.
A B C
P = π(0)= [1000, 1000, 1000)
Using the data given in Table 14-3, what is the long run number of employees expected in location C?

A)1400
B)1000
C)800
D)750

Question 73

Table 14-5
The following data consists of a matrix of transition probabilities (P)of three potential diseases, and the initial incidence of each disease π(0).Assume that each state represents a disease (Disease 1, Disease 2, Disease 3, respectively)and the transition probabilities represent changes from one checkup to the next.
P =

π(0)= (.3, .3, .4)
Using the data in Table 14-5, determine Disease 3's estimated incidence in the next period.

A)0.32
B)0.34
C)0.36
D)0.38

Question 74

Table 14-4
Cuthbert Wylinghauser is a scheduler of transportation for the state of Delirium.This state contains three cities: Chaos (C1), Frenzy (C2), and Tremor (C3).A transition matrix, indicating the probability that a resident in one city will travel to another, is given below.Cuthbert's job is to schedule the required number of seats, one to each person making the trip (transition), on a daily basis.
C F T
Transition matrix:

π(0)= [100, 100, 100]
Using the data given in Table 14-4, how many seats should Cuthbert schedule for travel from Chaos to Frenzy for tomorrow?

A)110
B)100
C)90
D)80

Question 75

Table 14-6
The following data consists of a matrix of transition probabilities (P)of four majors in the College of Business, and the initial proportion of students in each major π(0).Assume that each state represents a major and the transition probabilities represent changes from one major to the next after taking the introductory class in each discipline.
P =

π(0)= (.4, .3, .2, .1)
Using the data in Table 14-6, determine Major 3's estimated popularity after students have taken the first two introductory courses.

A)0.232
B)0.220
C)0.356
D)0.191

Question 76

Table 14-2
The following data consists of a matrix of transition probabilities (P)of three competing retailers, the initial market share π(0).Assume that each state represents a retailer (Retailer 1, Retailer 2, Retailer 3, respectively)and the transition probabilities represent changes from one month to the next.
P = π(0)= (0.3, 0.6, 0.1)
Using the data given in Table 14-2, find the market shares for the three retailers in month 2.

A)π(2)= (0.30, 0.60, 0.10)
B)π(2)= (0.55, 0.33, 0.12)
C)π(2)= (0.44, 0.43, 0.12)
D)π(2)= (0.55, 0.12, 0.33)

Question 77

Table 14-4
Cuthbert Wylinghauser is a scheduler of transportation for the state of Delirium.This state contains three cities: Chaos (C1), Frenzy (C2), and Tremor (C3).A transition matrix, indicating the probability that a resident in one city will travel to another, is given below.Cuthbert's job is to schedule the required number of seats, one to each person making the trip (transition), on a daily basis.
C F T
Transition matrix:

π(0)= [100, 100, 100]
Using the data given in Table 14-4, what is the equilibrium travel population of Chaos (rounded to the nearest whole person)?

A)79
B)95
C)126
D)100

Question 78

Table 14-3
The following data consists of a matrix of transition probabilities (P)of three office locations (A,B,C)within a large company and how employees shift from one location to the other from year to year.The company CEO would like to understand the movement of employees over time and the long-run proportion of employees in each location.Assume that there is always a total of 3000 employees.
A B C
P = π(0)= [1000, 1000, 1000)
Using the data given in Table 14-3, how many employees do we expect in location A two years from now?

A)1000
B)1400
C)1420
D)1500

Question 79

Table 14-6
The following data consists of a matrix of transition probabilities (P)of four majors in the College of Business, and the initial proportion of students in each major π(0).Assume that each state represents a major and the transition probabilities represent changes from one major to the next after taking the introductory class in each discipline.
P =

π(0)= (.4, .3, .2, .1)
Using the data in Table 14-6, determine Major 4's estimated popularity after students have taken the first two introductory courses.

A)0.232
B)0.220
C)0.356
D)0.191

Question 80

Table 14-6
The following data consists of a matrix of transition probabilities (P)of four majors in the College of Business, and the initial proportion of students in each major π(0).Assume that each state represents a major and the transition probabilities represent changes from one major to the next after taking the introductory class in each discipline.
P =

π(0)= (.4, .3, .2, .1)
Using the data in Table 14-6, determine Major 2's estimated popularity after students have taken the first introductory course.

A)0.225
B)0.305
C)0.245
D)0.205