![1. [20 pts.] Consider the Markov chains, MC-1 and MC-2, shown below. MC-1 MC-2 1. 10 pes Camisa de Markovering med and Mc.2,](http://img.homeworklib.com/questions/cabd0220-6d12-11ec-9d8c-a30de9744594.png?x-oss-process=image/resize,w_560)
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1. [20 pts.] Consider the Markov chains, MC-1 and MC-2, shown below. MC-1 MC-2 1. 10...
Markov Chains Consider the Markov chain with transition matrix P = [ 0 1 1 0]....
Markov Chains Consider the Markov chain with transition matrix P = [ 0 1 1 0]. 1) Compute several powers of P by hand. What do you notice? 2) Argue that a Markov chain with P as its transition matrix cannot stabilize unless both initial probabilities are 1/2.
6. [10 pts.] Suppose a computer system works in two modes of operation: a sleep mode, A', where i...
6. [10 pts.] Suppose a computer system works in two modes of operation: a sleep mode, A', where it is under-utilized and working mode A where it is adequately utilized. Every hour the computer system changes its state according to the following state diagram: 0.2 0.1 a. Finish labeling the transition on the state diagram. b. Give the corresponding probability matrix. c. What are the probabilities of being in the states A and A' after 3 hours of work. Suppose...
Consider the Markov chains given by the following transition matrices. (1) Q = (1/2 1/2) (we=...
Consider the Markov chains given by the following transition matrices. (1) Q = (1/2 1/2) (we= (1/2 162) (ii) Q = (1 o). /1/3 0 2/3 (1/2 1/2 0 (iv) Q = 1 0 1 0 1 (v) Q = 1 0 1/2 1/2 lo 1/5 4/5) \1/3 1/3 1/3) For each of the Markov chains above: A. Draw the transition diagram. B. Determine whether the chain is reducible or irreducible. Justify your answer. C. Determine whether the chain is...
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....
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....
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...
6. In the Markov Chain (MC) shown in Fig. 2, the two transitions out of any...
6. In the Markov Chain (MC) shown in Fig. 2, the two transitions out of any given state take place with equal probability (i.e., probability equal to ) (a) Write down a probability transition matrix P for this MC (b) Identify a stationary distribution q for this MC Note: Any solution toP with all 20, is termed as a stationary distribution. (c) Identify if possible, a steady-state probability vector r for the MC. Figure 2: A four-state Markov Chain. (Source:...
Let [An be a Markov chain (MC) wi ith the state space 1, 1/3 1/6 1/2...
Let [An be a Markov chain (MC) wi ith the state space 1, 1/3 1/6 1/2 1/3 1/3 1/3 2,31, transition matrix P- 0 1/4 3/4 and initial distribution o (1/3, 1/6, 1/2) (a) Draw the transition diagram (c) P(X 2, X2 3, X 1
Markov Chains: Consider the following transition matrix. Current month Next month Card used Card not used...
Markov Chains: Consider the following transition matrix. Current month Next month Card used Card not used .8 .2 .3 Card used Card not used The columns give probabilities a credit card will be used in the next month given that is used or not used in the current month (represented by rows). For example, the probability that a credit card is used next month, given that it was used in the current month is .8 or 80%. And, for example,...
6. [10 pts.] Suppose a computer system works in two modes of operation: a sleep mode, A', where it is under-utilized and working mode A where it is adequately utilized. Every hour the computer system changes its state according to the following state diagram: 0.2 0.1 a. Finish labeling the transition on the state diagram. b. Give the corresponding probability matrix. c. What are the probabilities of being in the states A and A' after 3 hours of work. Suppose...
Consider the Markov chains given by the following transition matrices. (1) Q = (1/2 1/2) (we= (1/2 162) (ii) Q = (1 o). /1/3 0 2/3 (1/2 1/2 0 (iv) Q = 1 0 1 0 1 (v) Q = 1 0 1/2 1/2 lo 1/5 4/5) \1/3 1/3 1/3) For each of the Markov chains above: A. Draw the transition diagram. B. Determine whether the chain is reducible or irreducible. Justify your answer. C. Determine whether the chain is...
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....
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....
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...
6. In the Markov Chain (MC) shown in Fig. 2, the two transitions out of any given state take place with equal probability (i.e., probability equal to ) (a) Write down a probability transition matrix P for this MC (b) Identify a stationary distribution q for this MC Note: Any solution toP with all 20, is termed as a stationary distribution. (c) Identify if possible, a steady-state probability vector r for the MC. Figure 2: A four-state Markov Chain. (Source:...
Let [An be a Markov chain (MC) wi ith the state space 1, 1/3 1/6 1/2 1/3 1/3 1/3 2,31, transition matrix P- 0 1/4 3/4 and initial distribution o (1/3, 1/6, 1/2) (a) Draw the transition diagram (c) P(X 2, X2 3, X 1
Markov Chains: Consider the following transition matrix. Current month Next month Card used Card not used .8 .2 .3 Card used Card not used The columns give probabilities a credit card will be used in the next month given that is used or not used in the current month (represented by rows). For example, the probability that a credit card is used next month, given that it was used in the current month is .8 or 80%. And, for example,...
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