Stochastic Processes: 1. What does it mean for a Markov Chain to be irreducible? 2. What...

Stochastic Processes:

1. What does it mean for a Markov Chain to be irreducible?

2. What simple conditions imply that a Markov Chain is irreducible?

Expert Solution

Colby Messinger answered 2 years ago

Is the following statement always true? “If an irreducible Markov chain has period 2, then for...

Is the following statement always true? “If an irreducible Markov chain has period 2, then for every state i∈S we have P2ii>0.” (Prove if “yes”, provide a counterexample if “no”)

For an irreducible Markov chain, either all states are positive recurrent or none are. Prove.

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The following is the transition probability matrix of a Markov chain with states 1, 2, 3,...

The following is the transition probability matrix of a Markov chain with states 1, 2, 3, 4 P 0 1 2 3 0 .4 .3 .2 .1 1 .2 .2 .2 .4 2 .25 .25 .5 0 3 .2 .1 .4 .3 If Xnot = 1 (a) find the probability that state 3 is entered before state 4; (b) find the mean number of transitions until either state 3 or state 4 is entered.

Let Xn be the Markov chain with states S = {1, 2, 3, 4} and transition...

Let Xn be the Markov chain with states S = {1, 2, 3, 4} and transition matrix. 1/3 2/3 0 0 2/3 0 1/3 0 1/3 1/3 0 1/3 0 1/3 2/3 0 a.) Let X0 = 3 and let T3 be the first time that the Markov chain returns 3, compute P(T3 = 2 given X0=3). Please show all work and all steps. b.) Find the stationary distribution π. Please show all work and all steps.

Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2,...

Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3}, and transition probability matrix (pij ) given by   2 3 1 3 0 0 1 3 2 3 0 0 0 1 4 1 4 1 2 0 0 1 2 1 2   Determine all recurrent states. Q3. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2} and transition probability matrix (pij...

Xn is a Markov Chain with state-space E = {0, 1, 2}, and transition matrix 0.4...

Xn is a Markov Chain with state-space E = {0, 1, 2}, and transition matrix 0.4 0.2 0.4 P = 0.6 0.3 0.1 0.5 0.3 0.2 And initial probability vector a = [0.2, 0.3, 0.5] For the Markov Chain with state-space, initial vector, and transition matrix discuss how we would calculate the follow; explain in words how to calculate the question below. a) P(X1 = 0, X2 = 0, X3 = 1, X4 = 2|X0 = 2) b) P(X2 =...

Consider the following Markov chain: 0 1 2 3 0 0.3 0.5 0 0.2 1 0.5...

Consider the following Markov chain: 0 1 2 3 0 0.3 0.5 0 0.2 1 0.5 0.2 0.2 0.1 2 0.2 0.3 0.4 0.1 3 0.1 0.2 0.4 0.3 What is the probability that the first passage time from 2 to 1 is 3? What is the expected first passage time from 2 to 1? What is the expected first passage time from 2 to 2 (recurrence time for 2)? What is the relation between this expectation and the steady-state...

Xn is a discrete-time Markov chain with state-space {1,2,3}, transition matrix, P = .2 .1 .7...

Xn is a discrete-time Markov chain with state-space {1,2,3}, transition matrix, P = .2 .1 .7 .3 .3 .4 .6 .3 .1 a) find E[X1|X0=2] = b) The P(X9=1|X7=3) = C) The P(X2=2) =

1. What does Stanton mean when he refers to the “element of 2+2?” 2. What does...

1. What does Stanton mean when he refers to the “element of 2+2?” 2. What does Stanton say about temperament?

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