Deck 17: Markov Processes

Question 1

Analysis of a Markov process

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.

Answer 1

A

Question 2

The probability that the system is in state 2 in the 5th period is $\pi$ ₅(2).

Answer 2

False

Question 3

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.

Answer 3

False

Question 4

Markov processes use historical probabilities.

Question 5

At steady state

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

Question 6

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

Question 7

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

Question 8

A Markov chain cannot consist of all absorbing states.

Question 9

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

Question 10

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 the alternatives is correct.

Question 11

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 12

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 13

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 14

All Markov chains have steady-state probabilities.

Question 15

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.

Question 16

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 17

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 18

The fundamental matrix is used to calculate the probability of the process moving into each absorbing state.

Question 19

Absorbing state probabilities are the same as

A)steady state probabilities.
B)transition probabilities.
C)fundamental probabilities.
D)None of the alternatives is true.

Question 20

Steady state probabilities are independent of initial state.

Question 21

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 22

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 23

A state i is an absorbing state if p_ii = 0.

Question 24

Appointments in a medical office are scheduled every 15 minutes.Throughout the day,appointments will be running on time or late,depending on the previous appointment only,according to the following matrix of transition probabilities:

a.The day begins with the first appointment on time.What are the state probabilities for periods 1,2,3 and 4?
b.What are the steady state probabilities?

Question 25

Two airlines offer conveniently scheduled flights to the airport nearest your corporate headquarters.Historically,flights have been scheduled as reflected in this transition matrix.

a.If your last flight was on B,what is the probability your next flight will be on A?
b.If your last flight was on B,what is the probability your second next flight will be on A?
c.What are the steady state probabilities?

Question 26

Rent-To-Keep rents household furnishings by the month.At the end of a rental month a customer can: a)rent the item for another month,b)buy the item,or c)return the item.The matrix below describes the month-to-month transition probabilities for 32-inch stereo televisions the shop stocks.
What is the probability that a customer who rented a TV this month will eventually buy it?

Question 27

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 28

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

Question 29

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.

Question 30

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

a.What are the absorbing states?
b.What is the probability that a patient with symptom 2 will recover?

Question 31

The matrix of transition probabilities below deals with brand loyalty to Bark Bits and Canine Chow dog food.

a.What are the steady state probabilities?
b.What is the probability that a customer will switch brands on the next purchase after a large number of periods?

Question 32

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

Question 33

Transition probabilities are conditional probabilities.

Question 34

A city is served by three cable TV companies: Xcellent Cable,Your Cable,and Zephyr Cable.A survey of 1000 cable subscribers shows this breakdown of customers from the beginning to the end of August.

a.Construct the transition matrix.
b.What was each company's share of the market at the beginning and the end of the month?
c.If the current trend continues what will the market shares be?

Question 35

A television ratings company surveys 100 viewers on March 1 and April 1 to find what was being watched at 6:00 p.m.-- the local NBC affiliate's local news,the CBS affiliate's local news,or "Other" which includes all other channels and not watching TV.The results show

a.What are the numbers in each choice for April 1?
b.What is the transition matrix?
c.What ratings percentages do you predict for May 1?

Question 36

The sum of the probabilities in a transition matrix equals the number of rows in the matrix.

Question 37

Bark Bits Company is planning an advertising campaign to raise the brand loyalty of its customers to .80.
a.The former transition matrix is
.75 .25
.20 .80
What is the new one?
b.What are the new steady state probabilities?
c.If each point of market share increases profit by $15000,what is the most you would pay for the advertising?

Question 38

State j is an absorbing state if p_ij = 1.

Question 39

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?

Question 40

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

Question 41

Rent-To-Keep rents household furnishings by the month.At the end of a rental month a customer can: a)rent the item for another month,b)buy the item,or c)return the item.The matrix below describes the month-to-month transition probabilities for 32-inch stereo televisions the shop stocks.
Next Month
_{Rent Buy Return}
What is the probability that a customer who rented a TV this month will eventually buy it?

Question 42

A recent study done by an economist for the Small Business Administration investigated failures of small business.Failures were either classified as due to poor financing,poor management,or a poor product.The failure rates differed for new businesses (under one year old)versus established businesses (over one year old. )
As the result of the economist's study,the following probabilities were determined.For new businesses the probability of failure due to financing was .15,due to management .20,and due to product .05.The corresponding probabilities for established businesses were .10,.06,and .03 respectively.
a.Determine a five-state Markov Chain transition matrix with states for new,established,and each of the three failure states.Write it in the form of I,O,R,and Q submatrices.
b.Determine the probability that a new business will survive during the next three years.
c.What proportion of new businesses eventually fail due to:
(1)poor financing? (2)poor management? (3)poor product?

Question 43

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 .8.If it is congested,he will take route B the next day with probability .7.
On the other hand,if on a day he takes route B and it is not congested,he will take route B again the next day with probability .9.Similarly if route B is congested,he will take route A the next day with probability .6.
a.Construct the transition matrix for this problem.(HINT: There are 4 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 44

Henry,a persistent salesman,calls North's Hardware Store once a week hoping to speak with the store's buying agent,Shirley.If Shirley does not accept Henry's call this week,the probability she will do the same next week is .35.On the other hand,if she accepts Henry's call this week,the probability she will not do so next week is .20.
a.Construct the transition matrix for this problem.
b.How many times per year can Henry expect to talk to Shirley?
c.What is the probability Shirley will accept Henry's next two calls if she does not accept his call this week?
d.What is the probability of Shirley accepting exactly one of Henry's next two calls if she accepts his call this week?