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Chapter 9: Q.9.6 (page 412)
Compute the limiting probabilities for the model of Problem 9.4.
The limiting probabilities is
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A transition probability matrix is said to be doubly
stochastic if
for all states j = 0, 1, ... , M. Show that such a Markov chain is ergodic, then
j = 1/(M + 1), j = 0, 1, ... , M.
Prove that if X can take on any of n possible values with respective probabilities P1, ... ,Pn, then H(X) is maximized when Pi = 1/n, i = 1, ... , n. What is H(X) equal to in this case?
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A random variable can take on any of n possible values x1, ... , xn with respective probabilities p(xi), i = 1, ... , n. We shall attempt to determine the value of X by asking a series of questions, each of which can be answered 鈥測es鈥� or 鈥渘o.鈥� For instance, we may ask 鈥淚s X = x1?鈥� or 鈥淚s X equal to either x1 or x2 or x3?鈥� and so on. What can you say about the average number of such questions that you will need to ask to determine the value of X?
This problem refers to Example 2f.
(a) Verify that the proposed value of 蟺j satisfies the necessary equations.
(b) For any given molecule, what do you think is the (limiting) probability that it is in urn 1?
(c) Do you think that the events that molecule j, j 脷 1, is in urn 1 at a very large time would be (in the limit) independent?
(d) Explain why the limiting probabilities are as given.
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