Question 1

Find the stationary matrix for the transition matrix P =  Round the numbers in your answer to the nearest hundredth.

Accepted Answer

A)  [0.44 0.48 0.08] 
B)  [0.33 0.40 0.27] 
C)  [1 0 0] 
D)  [0.33 0.67 0]

Question 2

According to data collected during one year in a large metropolitan community, 30% of commuters used public transportation to get to work, and this rose by 4% the following year. This is modeled by the transition matrix  where P represents the percentage of people that use public transportation and  the percentage of people that do not.
What percentage of commuters in this community will use public transportation in the long run?

Accepted Answer

A)  50% 
B)  34% 
C)  30% 
D)  37.2%

Question 3

Fayetteville is experiencing a population movement out of the city to the suburbs. Currently 85% of the total population live in the city with the remaining 15% living in the suburbs. It has been shown that each year 7% of the city residents move to the suburbs, while only 1% of the suburb population move back to the city. Assuming population remains constant for both, what percent of the total will remain in the suburbs after 5 years. Express your answer rounded to hundredths of a percent.

Accepted Answer

A)  35.56% 
B)  60.28% 
C)  39.72% 
D)  64.44

Question 4

Given the transition matrix:

Find the probability of going from state C to state B in four trials.

Accepted Answer

A)  0.0396 
B)  0.756 
C)  0 
D)  0.0144

Question 5

Find the stationary matrix for the transition matrix P =  .

Accepted Answer

A)   
B)  [0.6 0.4] 
C)   
D)  [0.636 0.364]

Question 6

The probability that a car owner will become a car renter in five years is 0.03. The probability that a renter will become an owner in five years is 0.1. Suppose the proportions in the population are 64% owners (O), 35.5% renters (R) and .5% neither (N) with the following transition matrix.

Accepted Answer

A)  [0.666 0.333 0.001] 
B)  [0.94 0.879 0.001] 
C)  [0.64 0.355 0.005] 
D)  [0.94 0.06 0]

Question 7

The transition matrix for a Markov process is:

Find  .

Accepted Answer

A)   
B)   
C)   
D)

Question 8

A trailer rental company has rental and return facilities at both a north and south location in a city. Assume a trailer must be returned to one or the other of these locations. If a trailer is rented at the north location, the probability that it will be returned there is .6; if a trailer is rented at the south location, the probability it will be returned there is .65. Assume the company rents all of its trailers each day and each trailer is rented (and returned) only once a day. If the company starts with 50% of the trailers at each location, what is the expected distribution (in percentages) the next day?

Accepted Answer

A)  62.5% of the trailers at the north location; 37.5% of the trailers at the south location 
B)  47.5% of the trailers at the north location; 52.5% of the trailers at the south location 
C)  52.5% of the trailers at the north location; 47.5% of the trailers at the south location 
D)  37.5% of the trailers at the north location; 62.5% of the trailers at the south location

Question 9

Find the stationary matrix for the transition matrix P =  Round the numbers in your answer to the nearest hundredth.

Accepted Answer

A)  [0.23 0.17 0.60] 
B)  [0.29 0.29 0.42] 
C)  [0.26 0.44 0.30] 
D)  [0.17 0.29 0.54]

Question 10

Find the stationary matrix for the transition matrix P =  .

Accepted Answer

A)   
B)   
C)   
D)  [0.25 0.45 0.35]

Question 11

will end up in state A over the long run.

Accepted Answer

A)   
B)   
C)  1 
D)

Question 12

A small town has only two dry cleaners, Fast and Speedy. Fast hopes to increase its market share by conducting an extensive advertising campaign. The initial market share for Fast was 40% and 60% for Speedy. Solve the problem:
-Find the probability that a customer using Fast initially will use Fast for his second batch of clothes. Use the following transition matrix.

Accepted Answer

A)  0. 73 
B)  0. 29 
C)  0. 27 
D)  0. 71

Question 13

Find all absorbing states for the transition matrix, and indicate whether or not the matrix is that of an absorbing Markov chain.

Accepted Answer

A)  State 3 is absorbing; matrix is an absorbing Markov chain. 
B)  State 2 is absorbing; matrix is not an absorbing Markov chain. 
C)  State 3 is absorbing; matrix is not an absorbing Markov chain. 
D)  State 2 is absorbing; matrix is an absorbing Markov chain.

Question 14

Among patients who were improving on a certain day (in the critical care unit of a certain hospital), the following was determined to be true on the next day: 58% were still improving; 10% were stable; 7% were deteriorating; and 25% had been discharged. Among the patients who were stable on a certain day, the following was determined to be true on the next day: 34% were improving; 44% were still stable; 18% were deteriorating; 3% had been discharged; and 1% had died. Among the patients who were deteriorating on a certain day, the following was determined to be true on the next day: 13% were improving; 42% were stable; 41% were still deteriorating; none had been discharged; and 4% had died. What is the expected number of additional days that a patient, who is improving on that certain day, will spend in the critical care unit? Round your answer to the nearest hundredth.

Accepted Answer

A)  5.76 days 
B)  4.65 days 
C)  7.53 days 
D)  3.22 days

Question 15

Dublin is experiencing a population movement out of the city to the suburbs. Currently 85% of the total population live in the city with the remaining 15% living in the suburbs. It has been shown that each year 7% of the city residents move to the suburbs, while only 1% of the suburb population move back to the city. Assuming population remains constant for both, what percent of the total will remain in the city after 2 years. Express your answer rounded to hundredths of a percent.

Accepted Answer

A)  31.05% 
B)  26.14% 
C)  73.86% 
D)  79.2%

Question 16

Find a standard form for the absorbing Markov chain with the following transition matrix

Accepted Answer

A)   
B)   
C)   
D)

Question 17

Given the transition matrix:

Find  .

Accepted Answer

A)   
B)   
C)   
D)

Question 18

Rats are kept in a cage with two compartments (A and B). Rats in A move to B with probability 0.7. Rats in B move to A with probability 0.2. Find the long-term trend for rats in each compartment. Round numbers to the nearest thousandth.

Accepted Answer

A)   
B)   
C)   
D)

Question 19

Weather is classified as sunny or cloudy in a certain place. What are the long-term predictions for sunny and cloudy days? Round numbers to the nearest thousandths.

Accepted Answer

A)   
B)   
C)   
D)

Question 20

According to data collected during one year in a large metropolitan community, 30% of commuters used public transportation to get to work, and this rose by 4% the following year. This is modeled by the transition matrix  where P represents the percentage of people that use public transportation and  the percentage of people that do not. Let  Find  .

Accepted Answer

A)  [0.9 0.1] 
B)  [0.3 0.7] 
C)  [0.66 0.34] 
D)  [0.34 0.66]

Exam 9: Markov Chains

Find the stationary matrix for the transition matrix P = Round the numbers in your answer to the nearest hundredth.

Given the transition matrix: Find the probability of going from state C to state B in four trials.

Find the stationary matrix for the transition matrix P = .

The probability that a car owner will become a car renter in five years is 0.03. The probability that a renter will become an owner in five years is 0.1. Suppose the proportions in the population are 64% owners (O) , 35.5% renters (R) and .5% neither (N) with the following transition matrix.

The transition matrix for a Markov process is: Find .

Find the stationary matrix for the transition matrix P = Round the numbers in your answer to the nearest hundredth.

Find the stationary matrix for the transition matrix P = .

will end up in state A over the long run.

Find all absorbing states for the transition matrix, and indicate whether or not the matrix is that of an absorbing Markov chain.

Find a standard form for the absorbing Markov chain with the following transition matrix

Given the transition matrix: Find .

Rats are kept in a cage with two compartments (A and B) . Rats in A move to B with probability 0.7. Rats in B move to A with probability 0.2. Find the long-term trend for rats in each compartment. Round numbers to the nearest thousandth.

Weather is classified as sunny or cloudy in a certain place. What are the long-term predictions for sunny and cloudy days? Round numbers to the nearest thousandths.