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- Set up a matrix, similar to the matrices we used in this chapter,
that corresponds to this state diagram:
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- Draw a picture corresponding to this transition matrix:
- Look closely at C in your picture. What do you notice that is strange
about the way information flows near C? What effect do you think this might
have on the long-range distribution of matter in this system?
- Which of the following are transition matrices? Explain.
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-
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- Which of these situations can be modeled by a homogeneous
Markov chain? If they cannot be modeled by a Markov chain, explain why not.
- The picture represents the probability that a delivery truck that is
currently in region i (A, B, or C) will be in region j (A, B, or C) for
the next time period.
- The picture represents the probability that a person eating meal i
(H, S, or P) for lunch today will eat meal j (H, S, or P) for lunch
tomorrow. The letter H stands for hamburger, S stands for salad, and P
stands for pizza.
- Look at the questions in problem 4. Write a sample transition
matrix for the problem or problems that can be modeled using a Markov chain.
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Assume S is the transition matrix
- What is the probability of going from state A to state B in one step?
- What is the probability of going from state B to state C in exactly
two steps?
- What is the probability of going from state C to state A in exactly
three steps?
- Give the transition matrix, S2, for two steps (S2 would give the probabilities of going from state i to state j in exactly 2 steps).
- Give the transition matrix for three steps.
- Give the transition matrix for four steps.
- To what matrix do these transition matrices appear to converge after
a large number of steps? Your solution should be accurate to two decimal
places.
- A math teacher, not wanting to be predictable, decided to assign
homework based on probabilities. On the first day of class, she drew this
picture on the board to tell the students whether to expect a full
assignment, a partial assignment, or no assignment the next day.
- Construct and label the transition matrix that corresponds to this
drawing. Label it A.
- If students have a full assignment today, what is the probability
that they will have a full assignment again tomorrow?
- If students have no assignment today, what is the probability
that they will have no assignment again tomorrow?
- Today is Wednesday and students have a partial assignment. What is
the probability that they will have no homework on Friday?
- Matrix A is the transition matrix for one day. Find the transition
matrix for two days (for example, if today is Monday, what are the chances
of getting each kind of assignment on Wednesday?).
- Find the transition matrix for three days.
- If you have no homework this Friday, what is the is the probability
that you will have no homework next Friday (since we are only considering
school days, there are only 5 days in a week)? Give your answer accurate to
two decimal places.
- Find, to two decimal places, the matrix to which
matrix A would appear to converge after many days.
- Explain the meaning of your solution to problem 7h.
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