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(Sheldon Ross) Consider a process {X,, п : 0, 1, . ( 1, 2, 31, suppose ..1, which takes on the values aij n even, Pj nodd, where j-1 for i = 1, 2, 3. Is {X, | n > 0} a Markov chain ? If not, show...
Consider a process {Xn, n = 0,1, ... }, which takes on the values 0,1, or 2. Suppose P{Xn+1 = ||Xn = i, Xn-1 = in-1,..., X0 = i +0} = when n is even when n is odd where - P = - Phl = 1, i = 0,1,2. Is {Xn, n = 0,1,... } a time-homogeneous Markov chain? If not, then show how, by enlarging the state space, we may transform it into a time- homogeneous Markov chain.
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 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?...
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 =
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. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix 1/2 1/3 1/6 0 1/4 (a) (6 points) Sketch the associated state transition diagram (b) (10 points) Suppose the Markov chain starts in state 1. What is the probability that it is in state 3 after two steps? (c) (10 points) Caleulate the steady-state distribution (s) for states 1, 2, and 3, respee- tively 1. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix...
4. Consider an irreducible Markov chain with finite state space S = {0, 1, , (a) Starting at state i, what is the probability that it will ever visit state j? (i,j arbi trary (b) Suppose that Xjj iyi for al i. Let ai P(visit N before 0 start at i). Show uations that the r, satisfy, and show that Xi . H2nt: Derive a system of linear eq that xi- solves these equations
1. Let {Xn, n 2 0 be a Markov Chain with state space S. Show that for any n, m-1 and JAn+m, . . . , İn+1,in-1 , . . . , io є S.
please help urgently solve number 1 #1. For a markov chain (X(n): n = 0, 1, ..} with state space {0, 1, 2, ...} and transition probability matrix P = Pial, let po be the probability mass function of X(0); that is, pli) = P(x(0) =i}. Give an expression for the probabilty mass function of X(n):
Consider a Markov chain with state space S = {0, 1, 2, 3} and transition probability matrix P= (a) Starting from state 1, determine the mean time that the process spends in each transient state 1 and 2, separately, prior to absorption. (b) Determine the mean time to absorption starting from state 1. (c) Starting from state 1, determine the probability for the process to be absorbed in state 0. Which state is it then more likely for the process...