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Chapter 9: Q.9.6 (page 412)

Compute the limiting probabilities for the model of Problem 9.4.

Short Answer

Expert verified

The limiting probabilities isπ=120920920120

Step by step solution

01

Given Information 

We have given that 3 white and 3 black balls are distributed in two urns in which each urn contains 3 balls.

We need to find the limiting probability.

02

Simplify

To find the distribution π=limn→∞Pns0for every starting distribution s0, we are solving

π=πP

Which yield the equalities

localid="1648053603504" π=19π1π1=π0+49π2π2=+49π1+49π2+π3π3=+19π2

Solving this system and using ∑iPi=1we end up with distribution

π=120920920120

This is the required stationary and limit distribution.

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Most popular questions from this chapter

A transition probability matrix is said to be doubly

stochastic if

∑i=0MPij=1

for all states j = 0, 1, ... , M. Show that such a Markov chain is ergodic, then

j = 1/(M + 1), j = 0, 1, ... , M.

Suppose that 3 white and 3 black balls are distributed in two urns in such a way that each urn contains 3 balls. We say that the system is in state i if the first urn contains i white balls, i = 0, 1, 2, 3. At each stage, 1 ball is drawn from each urn and the ball drawn from the first urn is placed in the second, and conversely with the ball from the second urn. Let Xn denote the state of the system after the nth stage, and compute the transition probabilities of the Markov chain {Xn, n Ú 0}.

This problem refers to Example 2f.

(a) Verify that the proposed value of πj 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 Ú 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.

A coin having probability p = 2 3 of coming up heads is flipped 6 times. Compute the entropy of the outcome of this experiment.

On any given day, Buffy is either cheerful (c), so-so (s), or gloomy (g). If she is cheerful today, then she will be c, s, or g tomorrow with respective probabilities .7, .2, and .1. If she is so-so today, then she will be c, s, or g tomorrow with respective probabilities .4, .3, and .3. If she is gloomy today, then Buffy will be c, s, or g tomorrow with probabilities .2, .4, and .4. What proportion of time is Buffy cheerful?
See all solutions

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