Deck 14: Markov Analysis

Question 1

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

Answer 1

False

Question 2

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

Answer 2

True

Question 3

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.

Answer 3

False

Question 4

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

Question 5

(n + 1)= nP

Question 6

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

Question 7

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

Question 8

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

Question 9

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 10

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 11

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 12

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

Question 13

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

Question 14

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

Question 15

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

Question 16

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

Question 17

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

Question 18

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

Question 19

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

Question 20

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

Question 21

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.
E)None of the above

Question 22

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.
E)None of the above

Question 23

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.
E)The size and the makeup of the system do not change during the analysis.

Question 24

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

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

Question 25

Markov analysis might be effectively used for

A)market share analysis.
B)university enrollment predictions.
C)machine breakdowns.
D)bad debt prediction.
E)All of the above

Question 26

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

Question 27

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.
E)state of technology.

Question 28

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)All of the above
E)None of the above

Question 29

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.
E)take the inverse of the fundamental matrix.

Question 30

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.
E)None of the above

Question 31

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)All of the above
E)None of the above

Question 32

In Markov analysis,we also assume that the sates are

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

Question 33

A state that when entered,cannot be left is called

A)transient.
B)recurrent.
C)absorbing.
D)steady.
E)None of the above

Question 34

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 35

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

Question 36

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

Question 37

In a matrix of transition probabilities (where i equals the row number and j equals the column number),

A)each number represents the conditional probability of being in state j in the next period given that it is currently in the state of i.
B)each number represents the probability that if something is in state i,it will go to state j in the next period.
C)the number in row 3,column 3 represents the probability that something will remain in state 3 from one period to the next.
D)the probabilities are usually determined empirically.
E)All of the above

Question 38

A state probability when equilibrium has been reached is called

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

Question 39

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

Question 40

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)joint probability.
E)transition probability.

Question 41

In Markov analysis,the fundamental matrix

A)is necessary to find the equilibrium condition when there are absorbing states.
B)can be found but requires,in part,partitioning of the matrix of transition probabilities.
C)is equal to the inverse of the I minus B matrix.
D)is multiplied by the A matrix in order to find the probabilities that amounts in non-absorbing states will end up in absorbing states.
E)All of the above

Question 42

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.
E)None of the above

Question 43

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)0.7
D)0.8
E)0.3

Question 44

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)identity matrix.
D)equilibrium matrix.
E)absorbing matrix.

Question 45

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
E)None of the above

Question 46

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)
E)π(1)= (0.33,0.33,0.33)

Question 47

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.
E)None of the above

Question 48

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.
E)the fundamental matrix is the same as the matrix of transition probabilities.

Question 49

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
E)None of the above

Question 50

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)
E)π(2)= (0.47,0.40,0.13)

Question 51

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
E)None of the above

Question 52

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
E)None of the above

Question 53

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
E)None of the above

Question 54

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.
E)None of the above

Question 55

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
E)multiply the fundamental matrix by the A matrix

Question 56

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
E)None of the above

Question 57

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

A)0%.
B)25%.
C)50%.
D)75%.
E)100%.

Question 58

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

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

Question 59

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
E)None of the above

Question 60

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)(0.30,0.60,0.10)
B)(0.55,0.33,0.12)
C)(0.44,0.43,0.12
D)(0.55,0.12,0.33)
E)(0.47,0.40,0.13)

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 matix: 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
E)None of the above

Question 62

Given the following matrix of transition probabilities,find the equilibrium states.

Question 63

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
E)820

Question 64

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

A)80
B)70
C)20
D)60
E)None of the above

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 A one year from now?

A)1000
B)1400
C)1500
D)800
E)700

Question 66

A certain utility firm has noticed that a residential customer's bill for one month is dependent on the previous month's bill.The observations are summarized in the following transition matrix. The utility company would like to know the long-run probability that a customer's bill will increase,the probability the bill will stay the same,and the probability the bill will decrease.

Question 67

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
E)700

Question 68

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 matix: 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
E)None of the above

Question 69

The vector of state probabilities for period n is (0.4,0.6).The accompanying matrix of transition probabilities is: Calculate the vector of state probabilities for period n+1.

Question 70

There is a 30% chance that any current client of company A will switch to company B this year.There is a 40% chance that any client of company B will switch to company A this year.If these probabilities are stable over the years,and if company A has 500 clients and company B has 300 clients,
(a)how many clients will each company have next year?
(b)how many clients will each company have in two years?

Question 71

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
E)700

Question 72

There is a 60% chance that a customer without a smart phone will buy one this year.There is a 95% chance that a customer with a smart phone will continue with a smart phone going into the next year.If 30% of target market currently own smart phones,what proportion of the target market is expected to own a smart phone next year?

Question 73

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 matix: 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
E)None of the above

Question 74

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)1000
B)1400
C)1420
D)1500
E)820

Question 75

A certain firm has noticed that employees' salaries from year to year can be modeled by Markov analysis.The matrix of transition probabilities follows. (a)Set up the matrix of transition probabilities in the form: (b)Determine the fundamental matrix for this problem.
(c)What is the probability that an employee who has received a raise will eventually quit?
(d)What is the probability that an employee who has received a raise will eventually be fired?

Question 76

The fax machine in an office is very unreliable.If it was working yesterday,there is an 90% chance it will work today.If it was not working yesterday,there is a 5% chance it will work today.
(a)What is the probability that it is not working today,if it was not working yesterday?
(b)If it was working yesterday,what is the probability that it is working today?

Question 77

Given the following vector of state probabilities and the accompanying matrix of transition probabilities,find the next period vector of state probabilities.
(0.4 0.4 0.2)

Question 78

There is a 60% chance that a customer without a smart phone will buy one this year.There is a 95% chance that a customer with a smart phone will continue with a smart phone going into the next year.If 30% of target market currently own smart phones,what is the long-run percentage of the target market that will own smart phones?

Question 79

There is a 30% chance that any current client of company A will switch to company B this year.There is a 20% chance that any client of company B will switch to company A this year.If these probabilities are stable over the years,and if company A has 1000 clients and company B has 1000 clients,in the long run (assuming the probabilities do not change),what will the market shares be?

Question 80

Over any given month,Hammond Market loses 10% of its customers to Otro Plaza and 20% to Tres Place.Otro Plaza loses 5% to Hammond and 10% to Tres Place.Tres Place loses 5% of its customers to each of the two competitors.At the present time,Hammond Market has 40% of the market,while the others have 30% each.
(a)Next month,what will the market shares be for the three firms?
(b)In two months,what will the market shares be for the three firms?