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...
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:...
Write down the most general transition matrix for a two state Markov chain (i.e. a random process that is Markov and homogenous). Prove that every such chain has an equilibrium vector. Classify the chains into those that are regular, absorbing and irreducible. Describe the general aysmptotic behavior in time of the chain when started from an arbitrary probability mass vector.
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.
P is the (one-step) transition probability matrix of a Markov chain with state space {0, 1, 2, 3, 4 0.5 0.0 0.5 0.0 0.0 0.25 0.5 0.25 0.0 0.0 P=10.5 0.0 0.5 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.0 0.0 0.0 0.5 0.5/ (a) Draw a transition diagram. (b) Suppose the chain starts at time 0 in state 2. That is, Xo 2. Find E Xi (c)Suppose the chain starts at time 0 in any of the states with...
Consider a Markov Chain on {1,2,3} with the given transition matrix P. Use two methods to find the probability that in the long run, the chain is in state 1. First, raise P to a high power. Then directly compute the steady-state vector. 3 P= 1 3 2 1 1 3 4 Calculate P100 p100 0.20833 0.20833 0.20833 0.58333 0.58333 0.58333 0.20833 0.20833 0.20833 (Type an integer or decimal for each matrix element. Round to five decimal places as needed.)...
1. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix 1/2 1/3 1/6 0 1/4 (a) (6 points) Sketch the associated state transition diagram (b) (10 points) Suppose the Markov chain starts in state 1. What is the probability that it is in state 3 after two steps? (c) (10 points) Caleulate the steady-state distribution (s) for states 1, 2, and 3, respee- tively 1. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix...
(n)," 2 0) be the two-state Markov chain on states (. i} with transition probability matrix 0.7 0.3 0.4 0.6 Find P(X(2) 0 and X(5) X() 0)
Let Xn be a Markov chain with state space {0, 1, 2}, and transition probability matrix and initial distribution π = (0.2, 0.5, 0.3). Calculate P(X1 = 2) and P(X3 = 2|X0 = 0) 0.3 0.1 0.6 p0.4 0.4 0.2 0.1 0.7 0.2
Consider a Markov chain {YmJnEN with state space S ,2, 3, 4), initial distribution Po (0.25,0.25, 0.5,0), and transition matrix 1/3 2/3 00 1/6 1/2 1/3 0 0 4/9 4/9 1/9 0 0 5/6 1/6 (a) Find the equilibrium probability distribution π (b) Find the probability PO1 3. Уг 3.Ý, 1) Consider a Markov chain {YmJnEN with state space S ,2, 3, 4), initial distribution Po (0.25,0.25, 0.5,0), and transition matrix 1/3 2/3 00 1/6 1/2 1/3 0 0 4/9...
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...