4. Let X be a Markov chain with state space S = {1, 2, 3) and transition matrix(²where 0 < p < 1. Prove thatP =0Ph =PPP1-P00P1-pain a2na3n alna2n a3nwhere ain + wa2n + w²a3n = (1 − p + pw)", w being a complex cube root of 1.a3na2nain

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Answered: 4. Let X be a Markov chain with state…

MATLAB: An Introduction with Applications

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Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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7. A transition matrix P is said to be doubly stochastic if the sum over each column equals 1: that is M Pij = 1, for all j. If such a chain is irreducible, aperiodic, and consists of M + 1 states, then 1 for j = 0, 1, ···, M. (a) Prove this result. Ij = M +1'
5. (i) Let 1 be a permutation on {1,2, ...,n}. The matrix P for which pi. = em(i). is called the permutation matrix associated with 7, where Pie is the i-th row of Pr and e;j = 1 if i = j and 0 otherwise. Let n = 4 and T = (3 2 4 1). Find Pr. (ii) Find the corresponding permutation matrix for the permutation %3D 1 2 3 4 3 4 1 2
A Markov chain {x}=0,1,2,.. satisfies the difference equation X = Axk-1 for every k ≥ 1, where - (0.9 A = {) 0.8 0.6 0.2 0.4 (i) Find the general term x for k≥ 1. (ii) What happens to x as k→ ∞o? - (0.3). and Xo =
• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain starts in state x, Pμ(A) the probability when the initial state is random with distribution μ. Ty = min{n ≥ 1: X₂ = y} is the first time after 0 that the chain visits state y. Px,y = Px(Ty < ∞) . Ny is the number of visits to state y after time 0. 4. Let Sn, n20 be a random walk on Z with step distribution 1-p 2 P 1) = 1/², P(X₁ = 1) = 2/₁ 2' 2' P(X₂ = 0) P(X₂ = − 1) = for some 0 < p < 1, p ‡ / . We may denote q=1 - p. That is, the increments (X₂)₁ are i.i.d. and S₂ = X₁ + … + Xn for n ≥ 1 and S = 0. Compute E[S2+1 Sn] for n ≥ 1. (b) Show that Mn = (q/p) Sn defines a martingale (with respect to (Xk)k-1). (c) Does the limit limn→∞ Mn exist almost surely? If yes, give a justification. If your answer is no, explain why. (d) Let T be the first time that S is equal to either −3 or 3. Compute P(ST = 3). Hint: You may use, without proof, the fact P(T<∞) = 1 and that the Optional Stopping Theorem…
Suppose qo = [1/4, 3/4] is the initial state distribution for a Markov process with the following transition matrix: %3D [1/2 1/2] M = [3/4 1/4] (a) Find q1, 92, and q3. (b) Find the vector v that qo M" approaches. (c) Find the matrix that M" approaches.
The sequence (Xn)n>o is a Markov chain with transition matrix 0 을 0 0 0 3 1 0 0 0 4 4 0 } 0 0 0 1 2 1 1 0 0 3 3 0 } 0 } 0 0 0 0 0 0 0 (a) Draw the transition diagram for (Xn). (b) Determine the communicating classes of the chain, stating whether each class is recurrent or transient. (c) Determine the period of each communicating class.
A Markov source S = {s1, s2, s3} has transition matrix M and equilibrium vector p: to to M = p = to 3 to 61 147 Calculate the average codeword length LMfor the Markov Huffman code HuffM for S: 12 [State your answer as a fraction.]
2. Consider a Markov chain {Xn}n≥o having the following transition diagram: 1/2 1 2 5 1/4 3 4 1/2 1/4 6 7 1/2 For this chain, there are two recurrent classes R₁ = {6,7} and R₂ = {1,2,5}, and one transient class R3 = {3,4}. (a) Find the period of state 3. (b) Find f33 and f22. (c) Starting at state 3, find the probability that the chain is absorbed into R₁. (d) Starting at state 3, find the mean absorbation time, i.e., the expected number of steps that the chain is absorbed into R₁ or R₂. Note: there are missing transition probabilities for this chain, but no impact for your solution.
2. For all permissible p values, determine the equivalence classes of the Markov chain with the following transition matrix P, classify states as transient or recurrent, and classify the Markov chain as irreducible or reducible. 0. 1-p 1 - 0. P = 0. 1-p 0. 1-p
22 Let X and Y be jointly Gaussian random variables where oy= oy and p =-1. Find a transformation matrix such that new random variables Y, and Y, are statistically independent.
A Markov system with two states satisfies the following rule. If you are in state 1 then of the time you change to state 2. If you are in state 2 then of the time you remain in state 2. At time t = 0, there are 100 people in state 1 and no people in the other state. state 1 Write the transition matrix for this system using the state vector v = state 2 T = Write the state vector for time t = 0. Vo Compute the state vectors for time t = 1 and t = 2. Vị = V2

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4. Let X be a Markov chain with state space S = {1, 2, 3) and transition matrix P (42) 0 0 where 0 < p < 1. Prove that P = P Ph = 0 P ain a2n a3n a3n aln a2n a2n a3n ain where ain + wa2n + w²a3n = (1 − p + pw)", w being a complex cube root of 1.