Homework Answers
(a) It is null recurrent.
(b) Transient.
(c) Transient.
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Consider the following random walk Cococoo -2 -1 2 1-p 1-p 1-p 1-p 1-p 1-p State whether the states of this Markov chai...
Q.5 6 marks Markov chain with the following (a) Draw the state transition diagram for transition matrix P 0 0.5 0 0.5 0...
Q.5 6 marks Markov chain with the following (a) Draw the state transition diagram for transition matrix P 0 0.5 0 0.5 0 0.2 0.8 0 0 O P = \ 0 0.1 0 0.2 0.7 0 0.9 0 0.1 0 0 0 0 0 1 on five states 1,2,3,4,5} 2 marks (b) Identify the communicating classes of the Markov chain and identify whether they are open or closed. Write them in set notation and mark them on the transition...
Consider the following Markov chain with the following transition diagram on states (1,2,3 2 1/3 1 1/4 2 3 s this Mar...
Consider the following Markov chain with the following transition diagram on states (1,2,3 2 1/3 1 1/4 2 3 s this Markov chain irreducible? 1 marks (a) (b) Find the probability of the Markov chain to move to state 3 after two time steps, providing it starts in state 2 [3 marks 14 Find the stationary distribution of this Markov chain [4 marks (c) (d) Is the stationary distribution also a limiting distribution for this Markov chain? Explain your answer...
2. The transition probabilities for several temporally homogeneous Markov chains with states 1,.,n appear below. For each: . Sketch a small graphical diagram of the chain (label the states and draw...
2. The transition probabilities for several temporally homogeneous Markov chains with states 1,.,n appear below. For each: . Sketch a small graphical diagram of the chain (label the states and draw the arrows, but you do not need to label the transition probabilities) . Determine whether there are any absorbing states, and, if so, list them. » List the communication classes for the chain . Classify the chain as irreducible or not . Classify each state as recurrent or transient....
1. Let Xn be a Markov chain with states S = {1, 2} and transition matrix...
1. Let Xn be a Markov chain with states S = {1, 2} and transition matrix ( 1/2 1/2 p= ( 1/3 2/3 (1) Compute P(X2 = 2|X0 = 1). (2) Compute P(T1 = n|Xo = 1) for n=1 and n > 2. (3) Compute P11 = P(T1 <0|Xo = 1). Is state 1 transient or recurrent? (4) Find the stationary distribution à for the Markov Chain Xn.
2. The transition probabilities for several temporally homogeneous Markov chains with states 1,.,n appear below. For each: . Sketch a small graphical diagram of the chain (label the states and draw...
2. The transition probabilities for several temporally homogeneous Markov chains with states 1,.,n appear below. For each: . Sketch a small graphical diagram of the chain (label the states and draw the arrows, but you do not need to label the transition probabilities) . Determine whether there are any absorbing states, and, if so, list them. » List the communication classes for the chain . Classify the chain as irreducible or not . Classify each state as recurrent or transient....
Problem 7.4 (10 points) A Markov chain Xo, X1, X2,.. with state space S = {1,2,3,4} has the following transition graph...
Problem 7.4 (10 points) A Markov chain Xo, X1, X2,.. with state space S = {1,2,3,4} has the following transition graph 0.5 0.5 0.5 0.5 0.5 0.5 2 0.5 0.5 (a) Provide the transition matrix for the Markov chain (b) Determine all recurrent and all transient states (c) Determine all communication classes. Is the Markov chain irreducible? (d) Find the stationary distribution (e) Can you say something about the limiting distribution of this Markov chain?
Problem 7.4 (10 points) A...
For the random walk of Example 4.18, use the strong law of large numbers to give another proof that the Markov chain is transient when p [Hint: Note that the state at time n can be written as Σίι Yi,...
For the random walk of Example 4.18, use the strong law of large numbers to give another proof that the Markov chain is transient when p [Hint: Note that the state at time n can be written as Σίι Yi, where the Y's are independent and PO-1)-p-1-PY,--1). Argue that if p 〉 흘, then, by the strong law of large numbers. Ση-1 Ý, oo as n oo and hence the initial state 0 can be visited only finitely often, and...
1. (15 points) For each of the following Markov Chains: specify the classes, determine whether they...
1. (15 points) For each of the following Markov Chains: specify the classes, determine whether they are transient or recurrent, draw state transition diagrams, find if any absorbent states, and write whether or not each of the chains is irreducible. (a) (5 points) 0.5 0.5 0 0 (b) (5 points) 2 0o P2=1 0 0 1 0 0 (c) (5 points) P3 = 4 2 4
Consider a three-state continuous-time Markov chain in which the transition rates are given by The states are labelled...
Consider a three-state continuous-time Markov chain in which the transition rates are given by The states are labelled 1, 2 and 3. (a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states. (b) Use the symmetry of Q to argue that this setting can be reduced to one with only 2 states. (c) Use the results of Problem 1 to solve the backward equations of...
Consider a three-state continuous-time Markov chain in which the transition rates are given by The states are labelled...
Consider a three-state continuous-time Markov chain in which the transition rates are given by The states are labelled 1, 2 and 3. (a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states. (b) Use the symmetry of Q to argue that this setting can be reduced to one with only 2 states. (c) Use the results of Problem 1 to solve the backward equations of...
Q.5 6 marks Markov chain with the following (a) Draw the state transition diagram for transition matrix P 0 0.5 0 0.5 0 0.2 0.8 0 0 O P = \ 0 0.1 0 0.2 0.7 0 0.9 0 0.1 0 0 0 0 0 1 on five states 1,2,3,4,5} 2 marks (b) Identify the communicating classes of the Markov chain and identify whether they are open or closed. Write them in set notation and mark them on the transition...
Consider the following Markov chain with the following transition diagram on states (1,2,3 2 1/3 1 1/4 2 3 s this Markov chain irreducible? 1 marks (a) (b) Find the probability of the Markov chain to move to state 3 after two time steps, providing it starts in state 2 [3 marks 14 Find the stationary distribution of this Markov chain [4 marks (c) (d) Is the stationary distribution also a limiting distribution for this Markov chain? Explain your answer...
2. The transition probabilities for several temporally homogeneous Markov chains with states 1,.,n appear below. For each: . Sketch a small graphical diagram of the chain (label the states and draw the arrows, but you do not need to label the transition probabilities) . Determine whether there are any absorbing states, and, if so, list them. » List the communication classes for the chain . Classify the chain as irreducible or not . Classify each state as recurrent or transient....
1. Let Xn be a Markov chain with states S = {1, 2} and transition matrix ( 1/2 1/2 p= ( 1/3 2/3 (1) Compute P(X2 = 2|X0 = 1). (2) Compute P(T1 = n|Xo = 1) for n=1 and n > 2. (3) Compute P11 = P(T1 <0|Xo = 1). Is state 1 transient or recurrent? (4) Find the stationary distribution à for the Markov Chain Xn.
2. The transition probabilities for several temporally homogeneous Markov chains with states 1,.,n appear below. For each: . Sketch a small graphical diagram of the chain (label the states and draw the arrows, but you do not need to label the transition probabilities) . Determine whether there are any absorbing states, and, if so, list them. » List the communication classes for the chain . Classify the chain as irreducible or not . Classify each state as recurrent or transient....
Problem 7.4 (10 points) A Markov chain Xo, X1, X2,.. with state space S = {1,2,3,4} has the following transition graph 0.5 0.5 0.5 0.5 0.5 0.5 2 0.5 0.5 (a) Provide the transition matrix for the Markov chain (b) Determine all recurrent and all transient states (c) Determine all communication classes. Is the Markov chain irreducible? (d) Find the stationary distribution (e) Can you say something about the limiting distribution of this Markov chain?
Problem 7.4 (10 points) A...
For the random walk of Example 4.18, use the strong law of large numbers to give another proof that the Markov chain is transient when p [Hint: Note that the state at time n can be written as Σίι Yi, where the Y's are independent and PO-1)-p-1-PY,--1). Argue that if p 〉 흘, then, by the strong law of large numbers. Ση-1 Ý, oo as n oo and hence the initial state 0 can be visited only finitely often, and...
1. (15 points) For each of the following Markov Chains: specify the classes, determine whether they are transient or recurrent, draw state transition diagrams, find if any absorbent states, and write whether or not each of the chains is irreducible. (a) (5 points) 0.5 0.5 0 0 (b) (5 points) 2 0o P2=1 0 0 1 0 0 (c) (5 points) P3 = 4 2 4
Consider a three-state continuous-time Markov chain in which the transition rates are given by The states are labelled 1, 2 and 3. (a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states. (b) Use the symmetry of Q to argue that this setting can be reduced to one with only 2 states. (c) Use the results of Problem 1 to solve the backward equations of...
Consider a three-state continuous-time Markov chain in which the transition rates are given by The states are labelled 1, 2 and 3. (a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states. (b) Use the symmetry of Q to argue that this setting can be reduced to one with only 2 states. (c) Use the results of Problem 1 to solve the backward equations of...
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