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 |
and initial probability vector a = [.2,.7,.1].
The P(X2=2) =
Xn is a discrete-time Markov chain with state-space {1,2,3}, transition matrix, P = .2 .1 .7...
Suppose that {Xn} is a Markov chain with state space S = {1, 2}, transition matrix (1/5 4/5 2/5 3/5), and initial distribution P (X0 = 1) = 3/4 and P (X0 = 2) = 1/4. Compute the following: (a) P(X3 =1|X1 =2) (b) P(X3 =1|X2 =1,X1 =1,X0 =2) (c) P(X2 =2) (d) P(X0 =1,X2 =1) (15 points) Suppose that {Xn} is a Markov chain with state space S = 1,2), transition matrix and initial distribution P(X0-1-3/4 and P(Xo,-2-1/4. Compute...
Let Xn be a Markov chain with state space {0,1,2}, the initial probability vector and one step transition matrix a. Compute. b. Compute. 3. Let X be a Markov chain with state space {0,1,2}, the initial probability vector - and one step transition matrix pt 0 Compute P-1, X, = 0, x, - 2), P(X, = 0) b. Compute P( -1| X, = 2), P(X, = 0 | X, = 1) _ a. 3. Let X be a Markov chain...
(10 points) Consider a Markov chain (Xn)n-0,1,2 probability matrix with state space S ,2,3) and transition 1/5 3/5 1/5 P-0 1/2 1/2 3/10 7/10 0 The initial distribution is given by (1/2,1/6,1/3). Compute (a) P[X2-k for all k- 1,2,3 (b) E[X2] Does the distribution of X2 computed in (a) depend on the initial distribution a? Does the expected value of X2 computed in (b) depend on the nitial distribution a? Give a reason for both of your answers.
Consider the Markov chain X0,X1,X2,... on the state space S = {0,1} with transition matrix P= (a) Show that the process defined by the pair Zn := (Xn−1,Xn), n ≥ 1, is a Markov chain on the state space consisting of four (pair) states: (0,0),(0,1),(1,0),(1,1). (b) Determine the transition probability matrix for the process Zn, n ≥ 1.
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
Suppose Xn is a Markov chain on the state space S with transition probability p. Let Yn be an independent copy of the Markov chain with transition probability p, and define Zn := (Xn, Yn). a) Prove that Zn is a Markov chain on the state space S_hat := S × S with transition probability p_hat : S_hat × S_hat → [0, 1] given by p_hat((x1, y1), (x2, y2)) := p(x1, x2)p(y1, y2). b) Prove that if π is a...
6. Suppose Xn is a two-state Markov chain with transition probabilities (Xn, Xn+1), n = 0, 1, 2, Write down the state space of the Markov chain Zo, Zi, . . . and determine the transition probability matrix.
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. 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.