# Quiz 12: Markov Process Models

Business

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Q 2Q 2

Although the number of possible states in a Markov process may be infinite, they must be countably infinite.

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Q 3Q 3

"How you arrived at where you are now has no bearing on where you go next." This, simply put, is the Markovian memoryless property.

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True

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Q 7Q 7

Markovian transition matrices are necessarily square.That is, there are exactly the same number of rows as there are columns.

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Q 11Q 11

For Markov processes with absorbing states, steady-state behavior is independent of the initial process state.

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Q 15Q 15

Markov processes are a powerful decision making tool useful in explaining the behavior of systems and determining limiting behavior.

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Q 16Q 16

All of the following are necessary characteristics of a Markov process except:
A)a countable number of stages.
B)a countable number of states per stage.
C)at least one absorbing state.
D)the memoryless property.

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Multiple Choice

Q 17Q 17

Which of the following is a necessary characteristic of a Markovian transition matrix?
A)Periodicity.
B)Column numbers sum to 1.
C)Square (number of rows = number of columns).
D)Singularity.

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Multiple Choice

Q 18Q 18

Consider the transition matrix: | .3 .2 .5 | | .1 .6 .3 |
| )2 .3 .5 |
The steady-state probability of being in state 1 is approximately:
A).177
B).231
C).300
D).403

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Multiple Choice

Q 19Q 19

The transition matrix | 0 1 0 | | 0 0 1 |
| 1 0 0 |
Represents what type of Markov process?
A)Periodic.
B)Absorbing.
C)Independent.
D)Nonrecurrent.

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Q 20Q 20

| 0.2 0.8 | | 0 1 | This transition matrix represents what type of Markov process?
A)Periodic.
B)Absorbing.
C)Independent.
D)Nonrecurrent.

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Multiple Choice

Q 21Q 21

Regarding a transition matrix which possesses an absorbing state:
A)All row values will not sum to 1.
B)There will be a complementary absorbing state.
C)At least two columns will be identical.
D)That state's row will consist of a "1" and "0's".

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Q 22Q 22

If a Markov process consists of two absorbing states and two nonabsorbing states, the limiting probabilities for the nonabsorbing states will:
A)both equal zero.
B)be 0.5 and 0.5.
C)be identical to the transient state probabilities.
D)depend on the state vector.

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Q 23Q 23

A state vector is used for determining the:
A)number of stages until steady-state is reached.
B)probability that the process is in a given state.
C)existence of absorbing states.
D)values of transient state probabilities.

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Multiple Choice

Q 24Q 24

Steady-state probabilities are independent of the initial state if:
A)the number of initial states is finite.
B)there are no absorbing states.
C)the number of states and stages are equal.
D)the process generates a fixed number of transient states.

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Multiple Choice

Q 25Q 25

A Markovian system is currently at stage 1.To determine the state of the system at stage 6, we must have, in addition to the transition matrix, the state probabilities at:
A)stage 5.
B)stage 1.
C)any stage, up to and including 5.
D)no stage values are needed.

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Multiple Choice

Q 26Q 26

The "mean recurrence time" for a state in a Markov process:
A)is the average time it takes to return to that given state.
B)is the complement of the steady-state value.
C)only applies to processes with absorbing states.
D)depends upon the total number of stages involved.

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Multiple Choice

Q 27Q 27

Retired people often return to the workforce.If a retired woman returns to work at the same place from which she retired -- even if only part time or for a limited term -- that signifies that retirement is not a:
A)transient state.
B)steady-state.
C)periodic state.
D)absorbing state.

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Multiple Choice

Q 28Q 28

A firm displeased with its projected steady-state market share may try to improve the situation by taking steps which hopefully will:
A)extend the number of stages.
B)alter the transition matrix.
C)better its transient state standing.
D)reduce the number of recurrent states.

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Multiple Choice

Q 29Q 29

A gambler has an opportunity to play a coin tossing game in which he wins his wager with probability .49 and loses his wager with probability .51.Suppose the gambler's initial stake is $40 and the gambler will continue to make $10 bets until his fortune either reaches $0 or $100 (at which time play will stop).Which of the following statements is true?
A)Increasing the amount of each wager from $10 to $20 will increase the expected playing time.
B)Increasing the initial stake to $50 will increase the expected playing time.
C)Reducing the initial stake to $20 will increase the expected playing time.
D)Increasing the probability of winning from .49 to 1.0 will increase the expected playing time.

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Multiple Choice

Q 30Q 30

In determining steady-state behavior for a process with absorbing states, the subdivision of the transition matrix yields:
A)an identity submatrix, but no zero submatrix.
B)no identity submatrix, but a zero submatrix.
C)both an identity submatrix and a zero submatrix.
D)neither an identity submatrix nor a zero submatrix.

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Multiple Choice

Q 31Q 31

The state vector for stage j of a Markov chain with n states:
A)is a 1 x n matrix.
B)contains transition probabilities for stage j.
C)contains only nonzero values.
D)contains the steady-state probabilities.

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Multiple Choice

Q 32Q 32

If we perform the calculations for steady-state probabilities for a Markov process with periodic behavior, what do we get?
A)Steady-state probabilities.
B)An unsolvable set of equations.
C)The fundamental matrix.
D)The long run percentage of time the process will be in each state.

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Multiple Choice

Q 33Q 33

What is the fundamental matrix for a Markov process with absorbing states?
A)A matrix composed of the identity submatrix, a zero submatrix, a submatrix of the transition probabilities between the non-absorbing states and the absorbing states, and a submatrix of transition probabilities between the non-absorbing states.
B)A matrix representing the average number of times the process visits the non-absorbing states.
C)The inverse of the identity matrix minus the matrix of the transition probabilities between the non-absorbing states and the absorbing states.
D)The matrix product of the limiting transition matrix and the matrix of transition probabilities between the non-absorbing states.

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Multiple Choice

Q 34Q 34

For a Markov process with absorbing states, we define Π(j) = state vector at stage j N = fundamental matrix
I = identity matrix
Q = matrix of transition probabilities between non-absorbing states
R = matrix of transition probabilities between non-absorbing states and absorbing states
The limiting state probabilities equal:
A)Π(1) * N * R
B)I * R * Q
C)(I - Q)

^{-1}D)Π(1) * N * QFree

Multiple Choice

Q 35Q 35

If we add up the values in the n rows of the fundamental matrix for a Markov process with absorbing states, what is the result?
A)The rows each add to 1.
B)The limiting probability for each state.
C)A meaningless number.
D)The mean time until absorption for each state.

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Q 38Q 38

Charles dines out twice a week.On Tuesdays, he always frequents the same Mexican restaurant; on Thursdays, he randomizes between Greek, Italian, or Thai (but never Mexican).Is this transient, periodic, or recurrent behavior?

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Q 43Q 43

When calculating steady-state probabilities, we multiply the vector of n unknown values times the transition probability matrix to produce n equations with n unknowns.Why do we arbitrarily drop one of these equations?

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Q 46Q 46

Every week a charter plane brings a group of high-stakes gamblers into Las Begas.Half the group stays and begins gambling at Hot Slots near the Strip, and the other half is housed and begins gambling at Better Bandits, some distance away.(Both hotel/casinos are owned by the same corporation.)

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Q 47Q 47

The transition matrix for customer purchases of alkaline batteries is believed to be as follows:

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Q 48Q 48

Suppose you play a coin flipping game with a friend in which a fair coin is used.If the coin comes up heads you win $1 from your

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Q 49Q 49

A simple computer game using a Markov chain starts at the Cave or the Castle with equal probability and ends with Death (you lose) or Treasure (you win).The outcome depends entirely on luck.The transition probabilities are:

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