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...
Question 4 9 marks Consider the process {XEN where Xn is the outcome of a die on the nth roll. (a) Show that {X,),EN Is a Markov chain. (b) Determine the state space S and the transition matrix P (with, as usual, reasoning). 0 3
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.
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...
Consider a Markov chain with state space S = {1,2,3,4} and transition matrix P = where (a) Draw a directed graph that represents the transition matrix for this Markov chain. (b) Compute the following probabilities: P(starting from state 1, the process reaches state 3 in exactly three-time steps); P(starting from state 1, the process reaches state 3 in exactly four-time steps); P(starting from state 1, the process reaches states higher than state 1 in exactly two-time steps). (c) If the...
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...
Q5. Consider a Markov chain {Xn|n ≥ 0} with state space S = {0, 1, · · · } and transition matrix (pij ). Find (in terms QA for appropriate A) P{ max 0≤k≤n Xk ≤ m|X0 = i} . Q6. (Flexible Manufacturing System). Consider a machine which can produce three types of parts. Let Xn denote the state of the machine in the nth time period [n, n + 1) which takes values in {0, 1, 2, 3}. Here...
Consider a Markov chain with state space S = {1, 2, 3, 4} and transition matrix P= where (a) Draw a directed graph that represents the transition matrix for this Markov chain. (b) Compute the following probabilities: P(starting from state 1, the process reaches state 3 in exactly three time steps); P(starting from state 1, the process reaches state 3 in exactly four time steps); P(starting from state 1, the process reaches states higher than state 1 in exactly two...
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...
(10 points) Consider a Markov chain (Xn)n-0,1,2 probability matrix with state space S ,2,3) and transition 1/5 3/5 1/5 P-0 1/2 1/2 3/10 7/10 0 The initial distribution is given by (1/2,1/6,1/3). Compute (a) P[X2-k for all k- 1,2,3 (b) E[X2] Does the distribution of X2 computed in (a) depend on the initial distribution a? Does the expected value of X2 computed in (b) depend on the nitial distribution a? Give a reason for both of your answers.
Xn is a discrete-time Markov chain with state-space {1,2,3}, transition matrix, P = .2 .1 .7 .3 .3 .4 .6 .3 .1 and initial probability vector a = [.2,.7,.1]. The P(X2=2) =