Question 1

Markov Analysis has many business applications, such as accounts receivables analysis and machine maintenance.

Accepted Answer

Markov Analysis is a statistical method that can be used to model and analyze various systems and processes, including accounts receivables to predict future cash flows and machine maintenance to forecast equipment failures, making it applicable in these business areas.

Question 2

Transition probabilities indicate the tendency of the system to change from one state to another after an elapse of one period.

Accepted Answer

Transition probabilities represent the likelihood of a system transitioning from one state to another in a given time period, often used in Markov models to predict future states based on current conditions.

Question 3

Short term behavior of a Markov system depends on the current state and transition probabilities.

Accepted Answer

In a Markov system, the short-term behavior (i.e., the next state) depends solely on the current state and the transition probabilities to other states, not on the sequence of events that preceded it.

Question 4

An absorbing state in a Markov system is one in which the system will get stuck and will not be able to get out of that state.

Accepted Answer

An absorbing state in a Markov chain is a state that, once entered, cannot be left. Hence, the system will remain in this state indefinitely once it is reached.

Question 5

Markov systems typically exist for one or two periods.

Accepted Answer

Markov systems, or Markov processes, do not have a limitation on the number of periods they can exist for; they can extend over an indefinite number of periods or states, depending on the specific system or process being modeled.

Question 6

In Markov systems, the probability of going from one state in period $n$ to another state in period $(n+1)$ depends on what states the system traveled in periods $1,2, \ldots$, n.

Accepted Answer

In Markov systems, the probability of transitioning from one state to another depends only on the current state, not on the sequence of states that preceded it.

Question 7

The states in a Markov system are mutually exclusive and collectively exhaustive.

Accepted Answer

In a Markov system, each state represents a unique condition or position in which the system can exist, and at any given time, the system must be in exactly one of these states, covering all possible scenarios without overlap.

Question 8

The columns of the transition probability matrix should add up to 1.0.

Accepted Answer

The rows of the transition probability matrix should add up to 1.0, not the columns. This ensures that the total probability of transitioning from one state to all possible states is 1.

Question 9

The rows of a transition probability matrix must add up to 1.0.

Accepted Answer

Each row in a transition probability matrix represents the probabilities of moving from one state to another, and since these probabilities must cover all possible outcomes, they sum up to 1.0.

Question 10

Short term behavior of a Markov system is determined by its transition probability matrix, while the long term behavior is dependent on the starting state of the system.

Accepted Answer

The long-term behavior of a Markov system is determined by its transition probability matrix and is independent of the starting state, assuming the system reaches a steady state or equilibrium distribution.

Question 11

A tree diagram is a very useful technique for analyzing the long term behavior of a Markov system.

Accepted Answer

A tree diagram is useful for representing all possible outcomes of a process or decision, but for analyzing the long-term behavior of a Markov system, a state transition matrix or a Markov chain diagram is more appropriate.

Question 12

If matrix $\mathrm{A}$ is multiplied by matrix $\mathrm{B}-(\mathrm{A} * \mathrm{~B})$ - then the number of rows in $\mathrm{A}$ should equal the number of columns in B.

Accepted Answer

For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix.

Question 13

Judy Jones purchases groceries and pop exactly once each week on Sunday evenings. She buys either Coke or Pepsi only and switches from Coke to Pepsi and vice-versa somewhat regularly. Her purchasing behavior of these two drinks is modeled as a Markov system. Querying Judy, a novice student came up with the following transition mattix. This matrix satisfies all the conditions for being a transition matrix.

Accepted Answer

The answer of Judy Jones purchases groceries and pop exactly...

Question 14

A transient state in a Markov chain is one in which the system cannot stay in for the long run.

Accepted Answer

A transient state in a Markov chain is one from which it is possible to exit and not return, meaning the system cannot remain in this state indefinitely as it transitions to other states.

Question 15

A state or a group of states in a Markov chain is called absorbing if the system can never get out of that state or the group of states (if it ever reaches it).

Accepted Answer

In a Markov chain, an absorbing state is one that, once entered, cannot be left. Similarly, a group of states is absorbing if, once the system enters any state within the group, it cannot exit to a state outside of that group.

Question 16

In Markov chains having absorbing states and transient states, in order to compute probability of absorption in an absorbing state starting from a transient state, the matrix is partitioned into two submatrices.

Accepted Answer

The answer of In Markov chains having absorbing states and...

Question 17

Inverse of a matrix exists even if the matrix in not a square matrix.

Accepted Answer

The inverse of a matrix exists only for square matrices, and even then, only if the matrix is non-singular (i.e., its determinant is not zero).

Question 18

If there are only two absorbing states and three transient states, we can add a dummy absorbing state to make the matrix I-Q into a square matrix.

Accepted Answer

Adding a dummy absorbing state makes the matrix I-Q into a square matrix because it increases the number of rows and columns by one, resulting in a square matrix.

Question 19

Newsweek and Time are two competing, weeklies each of which tries to keep their readership while at the same time trying to get the other's readers to switch. Among all the households holding yearly subscriptions to Newsweek or Time (but not both), let $\mathrm{N}$ and $\mathrm{T}$ denote the states that a household is a current Newsweek or Time subscriber, respectively. The probabilities of switching from one state to the other after one transition (when they renew) are given by the tree diagram. (For simplicity, it may be assumed that all annual subscriptions expire on December 31st and are renewed by all of them for one or the other but not both magazines for one year.) If the current number of households subscribing to Newsweek and Time is 2000 and 4000, respectively, Time would have a larger number of subscribers in the long run.

Accepted Answer

The answer of Newsweek and Time are two competing, weeklies...

Question 20

Newsweek and Time are two competing, weeklies each of which tries to keep their readership while at the same time trying to get the other's readers to switch. Among all the households holding yearly subscriptions to Newsweek or Time (but not both), let $\mathrm{N}$ and $\mathrm{T}$ denote the states that a household is a current Newsweek or Time subscriber, respectively. The probabilities of switching from one state to the other after one transition (when they renew) are given by the tree diagram. (For simplicity, it may be assumed that all annual subscriptions expire on December 31st and are renewed by all of them for one or the other but not both magazines for one year.) If the current number of households subscribing to Newsweek and Time is 8000 and 4000, respectively, Newsweek would have a larger number of subscribers in the long run.

Accepted Answer

The answer of Newsweek and Time are two competing, weeklies...

Quiz 12: Markov Analysis

Markov Analysis has many business applications, such as accounts receivables analysis and machine maintenance.

Transition probabilities indicate the tendency of the system to change from one state to another after an elapse of one period.

Short term behavior of a Markov system depends on the current state and transition probabilities.

An absorbing state in a Markov system is one in which the system will get stuck and will not be able to get out of that state.

Markov systems typically exist for one or two periods.

In Markov systems, the probability of going from one state in period $n$ to another state in period $(n+1)$ depends on what states the system traveled in periods $1,2, \ldots$ , n.

The states in a Markov system are mutually exclusive and collectively exhaustive.

The columns of the transition probability matrix should add up to 1.0.

The rows of a transition probability matrix must add up to 1.0.

Short term behavior of a Markov system is determined by its transition probability matrix, while the long term behavior is dependent on the starting state of the system.

A tree diagram is a very useful technique for analyzing the long term behavior of a Markov system.

If matrix $\mathrm{A}$ is multiplied by matrix $\mathrm{B}-(\mathrm{A} * \mathrm{~B})$ - then the number of rows in $\mathrm{A}$ should equal the number of columns in B.

A transient state in a Markov chain is one in which the system cannot stay in for the long run.

A state or a group of states in a Markov chain is called absorbing if the system can never get out of that state or the group of states (if it ever reaches it).

In Markov chains having absorbing states and transient states, in order to compute probability of absorption in an absorbing state starting from a transient state, the matrix is partitioned into two submatrices.

Inverse of a matrix exists even if the matrix in not a square matrix.

If there are only two absorbing states and three transient states, we can add a dummy absorbing state to make the matrix I-Q into a square matrix.

Quiz 12: Markov Analysis

Markov Analysis has many business applications, such as accounts receivables analysis and machine maintenance.

Transition probabilities indicate the tendency of the system to change from one state to another after an elapse of one period.

Short term behavior of a Markov system depends on the current state and transition probabilities.

An absorbing state in a Markov system is one in which the system will get stuck and will not be able to get out of that state.

Markov systems typically exist for one or two periods.

In Markov systems, the probability of going from one state in period nnn to another state in period (n+1)(n+1)(n+1) depends on what states the system traveled in periods 1,2,…1,2, \ldots1,2,… , n.

The states in a Markov system are mutually exclusive and collectively exhaustive.

The columns of the transition probability matrix should add up to 1.0.

The rows of a transition probability matrix must add up to 1.0.

Short term behavior of a Markov system is determined by its transition probability matrix, while the long term behavior is dependent on the starting state of the system.

A tree diagram is a very useful technique for analyzing the long term behavior of a Markov system.

If matrix A\mathrm{A}A is multiplied by matrix B−(A∗ B)\mathrm{B}-(\mathrm{A} * \mathrm{~B})B−(A∗ B) - then the number of rows in A\mathrm{A}A should equal the number of columns in B.

A transient state in a Markov chain is one in which the system cannot stay in for the long run.

A state or a group of states in a Markov chain is called absorbing if the system can never get out of that state or the group of states (if it ever reaches it).

In Markov chains having absorbing states and transient states, in order to compute probability of absorption in an absorbing state starting from a transient state, the matrix is partitioned into two submatrices.

Inverse of a matrix exists even if the matrix in not a square matrix.

If there are only two absorbing states and three transient states, we can add a dummy absorbing state to make the matrix I-Q into a square matrix.

In Markov systems, the probability of going from one state in period $n$ to another state in period $(n+1)$ depends on what states the system traveled in periods $1,2, \ldots$ , n.

If matrix $\mathrm{A}$ is multiplied by matrix $\mathrm{B}-(\mathrm{A} * \mathrm{~B})$ - then the number of rows in $\mathrm{A}$ should equal the number of columns in B.