Consider a Markov chain with transition matrix where 0< a, b,c <1. Find the stationary distribution.
Let Xo, X1,... be a Markov chain with transition matrix 1(0 1 0 P 2 0 0 1 for 0< p< 1. Let g be a function defined by g(x) =亻1, if x = 1, if x = 2.3. , Let Yn = g(x,), for n 0. Show that Yo, Xi, is not a Markov chain.
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 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-
1. A Markov chain {X,,n0 with state space S0,1,2 has transition probability matrix 0.1 0.3 0.6 P=10.5 0.2 0.3 0.4 0.2 0.4 If P(X0-0)-P(X0-1) evaluate P[X2< X4]. 0.4 and P 0-2) 0.2. find the distribution of X2 and
7. Define a Markov Chain on S-0,1,2,3,... with transition probabilities Pi,i+1 with 0<p < 1/2. Prove that the Markov Chain is reversible.
7. Define a Markov Chain on S = {0, 1, 2, 3,·.) with transition probabilities Po,1 1, pi,i+1 = 1-Pi,i-,-p, i 1 with 0<p < 1/2. Prove that the Markov Chain is reversible.
6. Define a Markov Chain on S- 10, 1,2, 3,...) with transition probabilities Po,1 1, with 0<p<1 (a) Is the MC irreducible? (b) For which values of p the Markov Chain is reversible?
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...
1.13. Consider the Markov chain with transition matrix: 1 0 0 0.1 0.9 2 0 0 0.6 0.4 3 0.8 0.2 0 0 4 0.4 0.6 0 0 (a) Compute p2. (b) Find the stationary distributions of p and all of the stationary distributions ofp2. (c) Find the limit of p2n(x, x) as n → oo.
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.