Consider a Markov chain with initial state vector Vo = and migration matrix [ 4 P=...
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...
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.
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.)...
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) =
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.
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...
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.
Consider a two state Markov chain with one-step transition matrix on the states 1,21, , 0<p+q<2. 91-9 ' Show, by induction or otherwise, that the n-step transition matrix is Ptg -99 Based upon the above equation, what is lim-x P(Xn-2K-1). How about limn→x P(Xn-
Consider the Markov chain with state space {0, 1,2} and transition matrix(a) Suppose Xo-0. Find the probability that X2 = 2. (b) Find the stationary distribution of the Markov chain
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...