6. Define a Markov Chain on S- 10, 1,2, 3,...) with transition probabilities Po,1 1, with...
Define a Markov Chain on S = {0, 1, 2, 3, . . .} with transition probabilities p0,1 = 1, pi,i+1 = 1 − pi,i−1 = p, i ≥ 1 with 0 < p < 1. (a) Is the MC irreducible? (b) For which values of p the Markov Chain is reversible? 6. Define a Markov Chain on S 0, 1,2, 3,...) with transition probabilities i>1 with 0<p<. (a) Is the MC irreducible? (b) For which values of p the...
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.
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.
(a) Is the MC irreducible? (b) For which values of p the Markov Chain is reversible? 6. Define a Markov Chain on S-10,1,2,3,...) with transition probabilities Po,1 pi,i+1 1 -pi,i-1 = p, i i>1 1 = with 0<p<
5. Define a Markov Chain on S {1, 2, 3, …} with transition probabilities pi,i+1- it 1 (a) Is the MC irreducible? (b) Are the states positive recurrent? (c) Find the invariant distribution.
5. Define a Markov Chain on S-1,2,3,..) with transition probabilities Pi i+1 (a) Is the MC irreducible? (b) Are the states positive recurrent? (c) Find the invariant distribution
Consider the Markov chain with state space S = {0,1,2,...} and transition probabilities I p, j=i+1 pſi,j) = { q, j=0 10, otherwise where p,q> 0 and p+q = 1.1 This example was discussed in class a few lectures ago; it counts the lengths of runs of heads in a sequence of independent coin tosses. 1) Show that the chain is irreducible.2 2) Find P.(To =n) for n=1,2,...3 What is the name of this distribution? 3) Is the chain recurrent?...
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
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.
2. The transition probabilities for several temporally homogeneous Markov chains with states 1,.,n appear below. For each: . Sketch a small graphical diagram of the chain (label the states and draw the arrows, but you do not need to label the transition probabilities) . Determine whether there are any absorbing states, and, if so, list them. » List the communication classes for the chain . Classify the chain as irreducible or not . Classify each state as recurrent or transient....