Let α and β be positive constants. Consider a continuous-time Markov chain X(t) with state space S = {0, 1, 2} and jump rates
q(i,i+1) = β for0≤i≤1
q(j,j−1) = α for1≤j≤2.
Find the stationary probability distribution π = (π0, π1, π2) for this chain.
Answer:
Given Data
Let α and β be positive constants
S = {0, 1, 2}
jump rates
X ( t ) be continuous time Markov chain with state space s = { 0 , 1 , 2 }
0 1 2
Stationary distribution is
****Please like it..
Let α and β be positive constants. Consider a continuous-time Markov chain X(t) with state space...
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...
1. Let (т, P) be a time-homogeneous discrete-time Markov chain with state space {1, . . . , (a) Show that the Markov chain is not stationary (i.e., SSS). (b) Suppose P is doubly stochastic and π- JJ, . . . , Đ. Then show that the Markov chain is stationary
Please give the detail solution to the problems. Let (T,P) be a time-homogeneous discrete-time Markov chain with state space {1, . . . ,J) (a) Show that the Markov chain is not stationary (i.e., SSS) (b) Suppose P is doubly stochastic and π = (1,7, . 1 . Then show that the Markov chain is stationary
1. Exit times. Let X be a discrete-time Markov chain (with discrete state space) and suppose pii > 0. Let T =min{n 21: X i} be the exit time from state i. Show that T has a geometric distribution with respect to the conditional probability P 1. Exit times. Let X be a discrete-time Markov chain (with discrete state space) and suppose pii > 0. Let T =min{n 21: X i} be the exit time from state i. Show that...
2. (10 points) Consider a continuous-time Markov chain with the transition rate matrix -4 2 2 Q 34 1 5 0 -5 (a) What is the expected amount of time spent in each state? (b) What is the transition probability matrix of the embedded discrete-time Markov chain? (c) Is this continuous-time Markov chain irreducible? (d) Compute the stationary distribution for the continuous-time Markov chain and the em- bedded discrete-time Markov chain and compare the two 2. (10 points) Consider a...
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
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.
Suppose that we have a finite irreducible Markov chain Xn with stationary distribution π on a state space S. (a) Consider the sequence of neighboring pairs, (X0, X1), (X1, X2), (X2, X3), . . . . Show that this is also a Markov chain and find the transition probabilities. (The state space will be S ×S = {(i,j) : i,j ∈ S} and the jumps are now of the form (i, j) → (k, l).) (b) Find the stationary distribution...
Consider a three-state continuous-time Markov chain in which the transition rates are given by The states are labelled 1, 2 and 3. (a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states. (b) Use the symmetry of Q to argue that this setting can be reduced to one with only 2 states. (c) Use the results of Problem 1 to solve the backward equations of...
Consider a three-state continuous-time Markov chain in which the transition rates are given by The states are labelled 1, 2 and 3. (a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states. (b) Use the symmetry of Q to argue that this setting can be reduced to one with only 2 states. (c) Use the results of Problem 1 to solve the backward equations of...