Let p E [0,1] with pメ, and let (Xn)n=o b l e the Markov chain on with initia [0,1] given by distribution δο and transi...
Got stuck on this problem for several hours, literally in a desperate situation, sincerely could any expert give a help? Many many thanks in advance!! Problem 4 (20p). Let p є 10, il with p , and let (Xn)n-0 be the Markov chain on Z with initial distribution 0 and transition matrix 11 : Z x Z O, j given by 1-p if y-r- 1 otherwise Use the strong law of large numbers to show that each state is transient....
Got stuck on this problem for several hours, literally in a desperate situation, sincerely could any expert give a help? Many many thanks in advance!! Problem 4 (20p). Let p є 10, il with p , and let (Xn)n-0 be the Markov chain on Z with initial distribution 0 and transition matrix 11 : Z x Z O, j given by 1-p if y-r- 1 otherwise Use the strong law of large numbers to show that each state is transient....
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 discrete Markov chain with transition matrix P . Show that the m-step transition probabilities are independent of the past. Hint: it is clear for m=1, apply mathematical induction on m
2. The Markov chain (Xn, n = 0,1, 2, ...) has state space S = {1, 2, 3, 4, 5} and transition matrix (0.2 0.8 0 0 0 0.3 0.7 0 0 0 P= 0 0.3 0.5 0.1 0.1 0.3 0 0.1 0.4 0.2 1 0 0 0 0 1 ) (a) Draw the transition diagram for this Markov chain.
Let X(n), n 0 be the two-state Markov chain on states (0,1) with transition probability matrix probability matrix 「1-5 Find: (a) P(x(1) = olX (0-0, X(2) = 0) (b) P(x(1)メx(2)). Note. (b) is an unconditional joint probability so you will nced t nclude the initi P(X(0-0)-To(0) and P(X(0-1)-n(0).
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...
For the random walk of Example 4.18, use the strong law of large numbers to give another proof that the Markov chain is transient when p [Hint: Note that the state at time n can be written as Σίι Yi, where the Y's are independent and PO-1)-p-1-PY,--1). Argue that if p 〉 흘, then, by the strong law of large numbers. Ση-1 Ý, oo as n oo and hence the initial state 0 can be visited only finitely often, and...
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 process E here Xn is the outcome of a die on the nth roll at XnEN is a Markov chain. (b) Determine the state space S and the transition matrix P (with, as usual, reasoning Consider the process E here Xn is the outcome of a die on the nth roll at XnEN is a Markov chain. (b) Determine the state space S and the transition matrix P (with, as usual, reasoning