2.30. Classify the states for the Markov chain with matrix P=1 .5 0 .25 0 0...
A4. Classify the states of the Markov chain with the following transition matrix. 0 3 0 1 Find the stationary distribution of each irreducible, recurrent subchain and hence obtain the mean recurrence time of each state. (8
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.
Question 5 9 marks Consider a Markov chain {YTheN with state space S = {1,2,3,4), initial distribution Po (0.25,0.25, 0.5,0), and transition matrix 1/3 2/3 0 0 p 1/6 1/2 1/30 0 4/9 4/9 1/9 0 0 5/6 1/6 2(a) Find the equilibrium probability distribution T (b) Find the probability P(-1%-3. Ya-1).
Question 5 9 marks Consider a Markov chain {YTheN with state space S = {1,2,3,4), initial distribution Po (0.25,0.25, 0.5,0), and transition matrix 1/3 2/3 0 0 p...
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.
2. Consider a Markov chain with state space S 1,2,3,4) with transition matrix 1/3 2/3 0 0 3/4 1/4 00 0 0 1/5 4/5 0 0 2/3 1/3, (a) (10 points) Is the Markov chain irreducible? Explain your answer ive three examples of stationary distributions.
Let P be the n*n transition matrix of a Markov chain with a finite state space S = {1, 2, ..., n}. Show that 7 is the stationary distribution of the Markov chain, i.e., P = , 2hTi = 1 if and only if (I – P+117) = 17 where I is the n*n identity matrix and 17 = [11...1) is a 1 * n row vector with all components being 1.
2. A Markov chain on states {0, 1, 2, 3, 4, 5} has transition probability matrix 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 Find all classes. Compute the limiting probabilities lim,o P5i for i = 0, 1, 2, 3,4, 5
2. A Markov chain on states {0, 1, 2, 3, 4, 5} has transition probability matrix 0 0 0 0 0 0 0 0 0...
Problem 5. A Markov chain Xn, n probability matrix: 0 with states 1, 2, 3 has the following transition 0 1/3 2/3 1/2 0 1/2 If P(o-: 1)-P(Xo-2-1/4, calculate E(%) (use a computer).
Problem 5. A Markov chain Xn, n probability matrix: 0 with states 1, 2, 3 has the following transition 0 1/3 2/3 1/2 0 1/2 If P(o-: 1)-P(Xo-2-1/4, calculate E(%) (use a computer).
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...
Consider a Markov chain with state space S = {1,2,3,4} and transition matrix P = where (a) Draw a directed graph that represents the transition matrix for this Markov chain. (b) Compute the following probabilities: P(starting from state 1, the process reaches state 3 in exactly three-time steps); P(starting from state 1, the process reaches state 3 in exactly four-time steps); P(starting from state 1, the process reaches states higher than state 1 in exactly two-time steps). (c) If the...