1. Suppose the state space of a Markov chain is {0, 1, 2, 3}. Its transition probabilitymatrix isP0123001/2 0 1/21001 0=2000131/2 0 0 1/2Recall f = P(Xn = 0, Xk 0, 1 ≤ k ≤ n − 1|X₁ = 0) for n ≥ 1.‡Compute (i) fo, fo, f3 and fo; and (ii) foto (which denotes Σx=1 fő).0000%00n=

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Answered: 1. Suppose the state space of a Markov…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Consider the Markov chain with state space S = {0, 1, 2,...} and transition probabili- ties Poo = 1 - P. P02 = P, and Pr,x+2= P₁ P2,2-1=1-p, for x = 1,2,... For which values of p is this a transient chain?
A Markov chain {s} with state space n={1, 2, 3} has a sequence of realizations and process transitions as follows: Transition (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3.1) (3,2) (3,3) t S: S+1 2 1 1. 1 1 1 1 1 3 1 3 3 3 4 3 1. 2 3 1. 2 1 7 2 2 1. 8 2 2 1 9 2 3 1. 10 3 11 1 1 12 1 1. 13 1 1. 14 1. 1. 15 3 1. 1. 16 1. Number of frequencies : 1 3. 2 2 1 1 1 1 a. determine the transition frequency matrix O= 2 12 93 021 022 °23 %3D 31 °32 °33) 1 P1 P12 P13 b. determine the estimated transition probability matrixP= 2 P21 P22 P23 3 P31 P32 P33)
[stochastic process1]
Recall that the graph of a discrete-time, finite-state, homogeneous Markov chain on two states 1, 2 with transition matrix is CO (: :) 1 T = If instead a new Markov chain on three states {1, 2, 3} has transition matrix 10 1 0 (a) Draw the corresponding graph for this new Markov chain. (b) Is the Markov chain with transition matrix T irreducible?
3. Consider a Markov chain on states {0, 1, 2, 3, 4, 5} with the transition probability matrix 0 1 2 3 4 5 0 1 00 1 0 2 3 4 000 38230 098910 ○ 147SOLCO 3418140 0 ○ ○ 1430 000 000 0 ст 5 00 001 Classify each state as transient or recurrent, and compute lim∞ P3 if it exists? 31
2. Consider a Markov chain on states {0, 1, 2, 3, 4,...} with the transition probability matrix 0 1 2 3 4 0 1 — Po Po 0 0 0 1 1 — P1 0 P1 0 0 2 1 P2 : 0 0 P2 0 ... : : Determine if the Markov chain in each case below is transient or recurrent. (i) Pn = e−1/(n+1) for n = 0, 1, 2, … · ·; (ii) Pn = e−1/(n+1)² for n = 0, 1, 2, . · ·
1. For each of the transition matrices given below, list absorbing state(s), and determine the period of each of the states in each Markov chain. (a) P = (d) P = 1120 CLI2L 1434L3 0 ܚ . ܟ ܝ. | ܟ ܝܕ . ܚ 0 00 0 0 0 0 0 3I4LI4 LIABI+ 0 0 (b) P = 1 0 Lo (e) P = 0 1 1 0 0 1333 COLITOLIN 0 0 0 0 0 213 100 0 0 3 00 100 0 -0 WIN 1 0 0 0 0 0 0 00 1 0 0 4|012 TWO O 0 0 0 0 0 (c) P = (f) P = 0 1|2 1|3 OLT20IN +15 O N|TN|TO 120 HINO OLI3 O 0 LI5L4L 1113 2I5
Please do the following question with handwritten working out
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. Let Zbe a Markov chain on a state space S = {1,… … , m} with transition probability matrix P(i, j) = Tij. Suppose that sample paths of Z are observed from which we record Njj #i →→ j transitions of Z, m N₁ = [N₁; (# transitions out of state i). j=1 It is known that the likelihood function of the observations of Z is given by m Nij ΠΠΠ. i=1_ j=1 L = Incorporating the constraint j = 1 for each i = S, j=1 m (a) Write the loglikelihood function of the observation. (b) Show that the maximum likelihood estimator of i → j transition probability ij is N = Tij = (c) Let m = 2 and assume that the number of transitions of Z is given by the matrix Nij N₁ 50 100 25 75 Use the above information to derive estimators for 11, 12, 721 and 22.

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