Deck 16: Markov Processes

Question 1

A unique matrix of transition probabilities should be developed for each customer.

Answer 1

False

Question 2

All Markov chains have steady-state probabilities.

Answer 2

False

Question 3

Steady-state probabilities are independent of initial state.

Answer 3

True

Question 4

A Markov chain cannot consist of all absorbing states.

Question 5

Transition probabilities are conditional probabilities.

Question 6

The probability that a system is in state 2 in the fifth period is π₅(2).

Question 7

All entries in a matrix of transition probabilities sum to 1.

Question 8

If a Markov chain has at least one absorbing state,steady-state probabilities cannot be calculated.

Question 9

A state i is a transient state if there exists a state j that is reachable from i,but the state i is not reachable from state j.

Question 10

All entries in a row of a matrix of transition probabilities sum to 1.

Question 11

A state is said to be absorbing if the probability of making a transition out of that state is zero.

Question 12

When absorbing states are present,each row of the transition matrix corresponding to an absorbing state will have a single 1 and all other probabilities will be 0.

Question 13

When absorbing states are present,each row of the transition matrix corresponding to an absorbing state will have a single 1 and all other probabilities will be 0.

Question 14

The fundamental matrix is derived from the matrix of transition probabilities and is relatively easy to compute for Markov processes with a small number of states.

Question 15

For Markov processes having the memoryless property,the prior states of the system must be considered in order to predict the future behavior of the system.

Question 16

Markov processes use historical probabilities.

Question 17

State j is an absorbing state if p_ij = 1.

Question 18

All Markov chain transition matrices have the same number of rows as columns.

Question 19

A state,i,is an absorbing state if,when i = j,p_ij = 1.

Question 20

If an absorbing state exists,then the probability that a unit will ultimately move into the absorbing state is given by the steady-state probability.

Question 21

Markov process models

A)study system evolution over repeated trials.
B)often study successive time periods.
C)are often used when the state of the system in any particular period cannot be determined with certainty.
D)All of these are correct.

Question 22

Markov process trials

A)are used to describe future behavior of the system.
B)are used to optimize the system.
C)lead to higher-order decision making.
D)All of these are correct.

Question 23

Absorbing state probabilities are the same as

A)steady-state probabilities.
B)transition probabilities.
C)fundamental probabilities.
D)None of these are correct.

Question 24

Precision Craft,Inc.,manufactures ornate pedestal sinks.On any day,the status of a given sink is either: (a)somewhere in the normal manufacturing process,(b)being reworked because of a detected flaw,(c)finished successfully,or (d)scrapped because a flaw could not be corrected.The transition matrix is as follows:
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a.What is the probability of a sink eventually being finished if it is currently in process?
b.What is the probability of a sink eventually being scrapped if it is currently in rework?
c.What is the probability that a sink currently in rework will have a "finished" status either tomorrow or the next day? (Hint: There are three ways this can happen.)

Question 25

The probability of reaching an absorbing state is given by the

A)R matrix.
B)NR matrix.
C)Q matrix.
D)(I − Q)−¹ matrix.

Question 26

A transition probability describes

A)the probability of a success in repeated,independent trials.
B)the probability a system in a particular state now will be in a specific state next period.
C)the probability of reaching an absorbing state.
D)None of these are correct.

Question 27

For a situation with weekly dining at either an Italian or Mexican restaurant,

A)the weekly visit is the trial and the restaurant is the state.
B)the weekly visit is the state and the restaurant is the trial.
C)the weekly visit is the trend and the restaurant is the transition.
D)the weekly visit is the transition and the restaurant is the trend.

Question 28

In Markov analysis,we are concerned with the probability that the

A)state is part of a system.
B)system is in a particular state at a given time.
C)time has reached a steady state.
D)transition will occur.

Question 29

Bark Bits Company is planning an advertising campaign to raise the brand loyalty of its customers to 0.80.
a.The former transition matrix is as follows:

What is the new one?
b.What are the new steady-state probabilities?
c.If each point of market share increases profit by $15,000,what is the most you would pay for the advertising?

Question 30

At steady state,

A)π₁(n + 1)> π₁(n).
B)π₁ = π_2.
C)π₁ + π₂ ≥ 1.
D)π₁(n + 1)= π_1.

Question 31

Transition probabilities indicate that a customer moves,or makes a transition,from a state in a given period to each state in the following period.

Question 32

On any particular day,an individual can take one of two routes to work.Route A has a 25% chance of being congested,whereas route B has a 40% chance of being congested.The probability of the individual taking a particular route depends on his previous day's experience.If one day he takes route A and it is not congested,he will take route A again the next day with probability 0.8.If it is congested,he will take route B the next day with probability 0.7.On the other hand,if he takes route B one day and it is not congested,he will take route B again the next day with probability 0.9.Similarly,if route B is congested,he will take route A the next day with probability 0.6.
a.Construct the transition matrix for this problem.(Hint: There are four states corresponding to the route taken and the congestion.The transition probabilities are products of the independent probabilities of congestion and next-day choice.)
b.What is the long-run proportion of time that route A is taken?

Question 33

The probability of going from state 1 in period 2 to state 4 in period 3 is

A)p₁₂.
B)p₂₃.
C)p₁₄.
D)p₄₃.

Question 34

The probability that a system is in a particular state after a large number of periods is

A)independent of the beginning state of the system.
B)dependent on the beginning state of the system.
C)equal to one half.
D)the same for every ending system.

Question 35

The probability of making a transition from state i in a given period to state j in the next period is denoted as

A)P_ij.
B)P = [i - j].
C)P( )ij.
D)None of these are correct.

Question 36

If the probability of making a transition from a state is 0,then that state is called a(n)

A)steady state.
B)final state.
C)origin state.
D)absorbing state.