5. Let (Xn)n be a Markov chain on a state space S with n-step transition probabilities...
Let X0,X1,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1 = in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following always hold: P(Xn ≥0|X0 ≥0,X1 ≥0,...,Xn−1 ≥0)=P(Xn ≥0|Xn−1 ≥0) ? (Prove if “yes”, provide a counterexample if “no”) Let Xo,Xi, be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X,-'n l Xo-io, Xi...
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...
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...
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...
Consider a Markov chain with transition probabilities p(x, y), with state space S = {1, 2, . . . , 10}, and assume X0 = 3. Express the conditional probability P3(X6 =7, X5 =3 | X4 =1, X9 =3) entirely in terms of (if necessary, multi-step) transition probabilities.
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.
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.
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
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.
5. Let X n 2 0} be a Markov chain with state space S = {0,1,2,...}. Suppose P{Xn+1 = 0|X,p = 0 3/4, P{Xn+1 = 1\Xn, P{Xn+1 = i - 1|X, 0 1/4 and for i > 0, P{X+1 = i + 1|X2 = i} i} 3/4. Compute the long run probabilities for this Markov chain = 1/4 and =