Question 1

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.

Accepted Answer

A) True 
 B)False  False

Question 2

Calculate the steady state probabilities for this transition matrix. $$\left[ \begin{array} { c c } 
.60 & .40 \
.30 & .70
\end{array} \right]$$

Accepted Answer

$\pi$₁ = 3/7, $\pi$₂ = 4/7

Question 3

The daily price of a farm commodity is up, down, or unchanged from the day before. Analysts predict that if the last price was down, there is a .5 probability the next will be down, and a .4 probability the price will be unchanged. If the last price was unchanged, there is a .35 probability it will be down and a .35 probability it will be up. For prices whose last movement was up, the probabilities of down, unchanged, and up are .1, .3, and .6.
a.Construct the matrix of transition probabilities.
b.Calculate the steady state probabilities.

Accepted Answer

a. $$\left[ \begin{array} { c c c } .50 & .40 & .10 \ .35 & .30 & .35 \ .10 & .30 & .60 \end{array} \right]$$ b.$\pi$₁ = .305,$\pi$₂ = .333, $\pi$₃ = .365

Question 4

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.

Accepted Answer

A) True 
 B)False

Question 5

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.

Accepted Answer

A) True 
 B)False

Question 6

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

Accepted Answer

A) True 
 B)False

Question 7

Analysis of a Markov process

Accepted Answer

A) describes future behavior of the system. 
B) optimizes the system. 
C) leads to higher order decision making. 
D) All of the alternatives are true. 
A) describes future behavior of the system. 
B) optimizes the system. 
C) leads to higher order decision making. 
D) All of the alternatives are true.

Question 8

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.

Accepted Answer

A) True 
 B)False

Question 9

At steady state

Accepted Answer

A)  $\pi$1(n+1) >  $\pi$1(n) 
B)  $\pi$1 =  $\pi$2 
C)  $\pi$1 +  $\pi$2 $\ge$ 1 
D)  $\pi$1(n+1) =  $\pi$1 
A)  $\pi$1(n+1) >  $\pi$1(n) 
B)  $\pi$1 =  $\pi$2 
C)  $\pi$1 +  $\pi$2 $\ge$ 1 
D)  $\pi$1(n+1) =  $\pi$1

Question 10

The medical prognosis for a patient with a certain disease is to recover, to die, to exhibit symptom 1, or to exhibit symptom 2. The matrix of transition probabilities is $$\begin{array} { c | c c c c } 
& \text { Recover } & \text { Die } & \mathrm { S } _ { 1 } & \mathrm {~S} _ { 2 } \
\hline \text { Recover } & 1 & 0 & 0 & 0 \
\text { Die } & 0 & 1 & 0 & 0 \
\mathrm {~S} _ { 1 } & 1 / 4 & 1 / 4 & 1 / 3 & 1 / 6 \
\mathrm {~S} _ { 2 } & 1 / 4 & 1 / 8 & 1 / 8 & 1 / 2
\end{array}$$ 
a.What are the absorbing states?
b.What is the probability that a patient with symptom 2 will recover?

Accepted Answer

a.States 1 and 2 (recover and

Question 11

Give two examples of how Markov analysis can aid decision making.

Accepted Answer

Markov analysis is a statistical techniq

Question 12

Markov processes use historical probabilities.

Accepted Answer

A) True 
 B)False

Question 13

Explain the concept of memorylessness.

Accepted Answer

The concept of memorylessness refers to

Question 14

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

Accepted Answer

A) True 
 B)False

Question 15

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

Accepted Answer

A) True 
 B)False

Question 16

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

Accepted Answer

A) True 
 B)False

Question 17

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

Accepted Answer

A) p12 
B) p23 
C) p14 
D) p43 
A) p12 
B) p23 
C) p14 
D) p43

Question 18

Accounts receivable have been grouped into the following states:
State 1:
Paid
State 2:
Bad debt
State 3:
0-30 days old
State 4:
31-60 days old
Sixty percent of all new bills are paid before they are 30 days old. The remainder of these go to state 4. Seventy percent of all 30 day old bills are paid before they become 60 days old. If not paid, they are permanently classified as bad debts.
a.Set up the one month Markov transition matrix.
b.What is the probability that an account in state 3 will be paid?

Accepted Answer

a. @#IMG-DLM& b.NR = @#IMG-DLM& So, the

Question 19

What assumptions are necessary for a Markov process to have stationary transition probabilities?

Accepted Answer

For a Markov process to have stationary

Question 20

A transition probability describes

Accepted Answer

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 the alternatives is correct. 
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 the alternatives is correct.

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.

Calculate the steady state probabilities for this transition matrix. $\left[ \begin{array} { c c } .60 & .40 \\.30 & .70\end{array} \right]$

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.

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.

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

Analysis of a Markov process

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.

At steady state

Give two examples of how Markov analysis can aid decision making.

Markov processes use historical probabilities.

Explain the concept of memorylessness.

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

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

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

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

What assumptions are necessary for a Markov process to have stationary transition probabilities?

A transition probability describes

Introduction

An Introduction to Linear Programming

Linear Programming: Sensitivity Analysis and Interpretation of Solution

Linear Programming Applications in Marketing, Finance and Operations Management

Advanced Linear Programming Applications

Distribution and Network Problems

Integer Linear Programming

Nonlinear Optimization Models

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Decision Analysis

Multicriteria Decisions

Forecasting

Linear Programming: Simplex Method

Simplex-Based Sensitivity Analysis and Duality

Solution Procedures for Transportation and Assignment Problems

Minimal Spanning Tree

Dynamic Programming

Filters

Exam 16: Markov Processes

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.

Calculate the steady state probabilities for this transition matrix. [.60.40.30.70]\left[ \begin{array} { c c } .60 & .40 \\.30 & .70\end{array} \right][.60.30​.40.70​]

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.

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.

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

Analysis of a Markov process

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.

At steady state

Give two examples of how Markov analysis can aid decision making.

Markov processes use historical probabilities.

Explain the concept of memorylessness.

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

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

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

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

What assumptions are necessary for a Markov process to have stationary transition probabilities?

A transition probability describes

Introduction

An Introduction to Linear Programming

Linear Programming: Sensitivity Analysis and Interpretation of Solution

Linear Programming Applications in Marketing, Finance and Operations Management

Advanced Linear Programming Applications

Distribution and Network Problems

Integer Linear Programming

Nonlinear Optimization Models

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Decision Analysis

Multicriteria Decisions

Forecasting

Linear Programming: Simplex Method

Simplex-Based Sensitivity Analysis and Duality

Solution Procedures for Transportation and Assignment Problems

Minimal Spanning Tree

Dynamic Programming

Filters

Calculate the steady state probabilities for this transition matrix. $\left[ \begin{array} { c c } .60 & .40 \\.30 & .70\end{array} \right]$