A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
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Textbook Question
Chapter 9, Problem 9.7PTE
A transition
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The transition matrix of a Markov chain is given by
0 0
P =
(a) Find two distinct stationary distributions of this Markov chain.
(b) Find the general form of the stationary distribution.
(c) If 70) = (, 4, ¿, 1, ) is the initial probability vector at time 0,
,π(m) = (금' 등,을 ,).
4'4'6' 6' 6
then show that limn
12
HIN O -160 2/3
HIN O 230116
O 0 -16 0 116
Let X be a random variable with sample space {1,2, 3} and probability distribu-
(G 1 ). Find a transition matrix P such that the Markov chain {X„}
tion T =
simulates X.
A Markov Chain has the transition matrix
r-[% *].
P =
and currently has state vector % % ]: What is the probability it will be in state 1
after two more stages (observations) of the process?
(A) %
(B) 0
(C) /2
(D) 24
(E) 12
(F) ¼
(G) 1
(H) 224
Chapter 9 Solutions
A First Course in Probability
Ch. 9 - Customers arrive at a bank at a Poisson rate ....Ch. 9 - Cars cross a certain point in the highway in...Ch. 9 - Suppose that in Problem 9.2, AI is agile enough to...Ch. 9 - Suppose that 3 white and 3 black balls are...Ch. 9 - Consider Example 2a. If there is a 50-50 chance of...Ch. 9 - Compute the limiting probabilities for the model...Ch. 9 - A transition probability matrix is said to be...Ch. 9 - On any given day, Buffy is either cheerful (c),...Ch. 9 - Suppose that whether it rains tomorrow depends on...Ch. 9 - A certain person goes for a run each morning. When...
Ch. 9 - Prob. 9.11PTECh. 9 - Determine the entropy of the sum that is obtained...Ch. 9 - Prove that if X can take on any of n possible...Ch. 9 - A pair of fair dice is rolled....Ch. 9 - A coin having probability p=23 of coming up heads...Ch. 9 - Prob. 9.16PTECh. 9 - Show that for any discrete random variable X and...Ch. 9 - Prob. 9.18PTECh. 9 - Events occur according to a Poisson process with...Ch. 9 - Prob. 9.2STPECh. 9 - Prob. 9.3STPECh. 9 - Prob. 9.4STPECh. 9 - Prob. 9.5STPE
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- 12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction randomly choose which way to go. Figure 3.28 (a) Construct the transition matrix for the Markov chain that models this situation. (b) Suppose we start with 15 robots at each junction. Find the steady state distribution of robots. (Assume that it takes each robot the same amount of time to travel between two adjacent junctions.)arrow_forwardExplain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.arrow_forwardGive me right solution according to the questionarrow_forward
- 2.- Let X₁, X₁,... be the Markov chain on state space {1,2,3,4} with transition matrix 1/2 1/2 0 0 1/7 0 3/7 3/7 1/3 1/3 1/3 0 0 2/3 1/6 1/6, (a) Explain how you can tell this Markov chain has a limiting distribution and how you could compute it. Your answer should refer to the relevant Theorems in the notes. (b) Find the limiting distribution for this Markov chain. (100) (c) Without doing any more calculations, what can you say about p₁,1 (100) ? and P2,1arrow_forwardExplain how you can tell this Markov chain has a limiting distribution and how you could compute it. Your answer should refer to the relevant Theorems.arrow_forwardPlease show all steps and explain clearly.arrow_forward
- choose correct option plzarrow_forwardLet Zm represent the outcome during the nth roll of a fair dice. Define the Markov chain X, to be the maximum outcome obtained so far after the nth roll, i.e., X, = max {Z1, Z2,..., Zn}. What is the transition probability p22 of the Markov chain {Xn}?arrow_forwardA continuous-time Markov chain (CTMC) has three states (1, 2, 3}. The average time the process stays in states 1, 2, and 3 are 2.1, 13.6, and 3.5 seconds, respectively. The steady-state probability that this CTMC is in the second state ( TT, ) isarrow_forward
- Let X be a Poisson(A) random variable. By applying Markov's inequality to the random variable W = etx, t > 0, show that P(X ≥ m) ≤ e-tmex(et-1). Hence show that, for m > >, e-^ (ex) m P(X ≥ m) ≤ mmarrow_forwardFind the limiting distribution for this Markov chain. Then give an interpretation of what the first entry of the distribution you found tells you based on the definition of a limiting distribution. Your answer should be written for a non-mathematician and should consist of between 1 and 3 complete sentences without mathematical symbols or terminology.arrow_forwardAn individual can contract a particular disease with probability 0.17. A sick person will recover dur- ing any particular time period with probability 0.44 (in which case they will be considered healthy at the beginning of the next time period). Assume that people do not develop resistance, so that pre- vious sickness does not influence the chances of contracting the disease again. Model as a Markov chain, give transition matrix on your paper. Find the probability that a healthy individual will be sick after two time periods.arrow_forward
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