2. Consider a Markov chain {Xn}n≥o having the following transition diagram:1/21251/4341/21/4671/2For this chain, there are two recurrent classes R₁ = {6,7} and R₂ = {1,2,5}, andone 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 numberof steps that the chain is absorbed into R₁ or R₂.Note: there are missing transition probabilities for this chain, but no impact foryour solution.

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Answered: 2. Consider a Markov chain {Xn}nzo…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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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)
• 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.
4
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.
Consider a continuous-time Markov chain whose jump chain is a random walk with reflecting barriers 0 and m where po,1 = 1 and pm,m-1 =1 and pii-1 = Pii+1 = for 1
a)
E5.2. Consider a Markov chain with state space S = {1,2,3, 4, 5, 6} and transition matrix 1 0 0 0 0 0 0 0 0 P = 0 0 0 1 0 0 0 0 0 0 0 1 0 (a) Draw the transition graph of this chain. (b) Show that state 1 is absorbing and conclude that it is positive re- current. (c) Show that states 2 and 3 are transient. (d) Verify that states 4, 5 and 6 are positive recurrent.
answer only e,f,g
3. Identify the null hypothesis, alternative hypothesis, test statistic, p-value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. The Genetics and IVF Institute conducted a clinical trial of the XSORT method designed to increase the probability of conceiving a girl. As this book was being written, 325 babies were born to parents using the XSORT method, and 295 of them were girls. Use the sample data with a 0.01 significance level to test the claim that with this method, the probability of a baby being a girl is greater than 0.5. Does the method appear to work?
1. Referring to Scenario 13-11, which of the following is the correct interpretation for the slope coefficient?a) For each decrease of 1 thousand downloads, the expected revenue is estimated to increase by $ 3.7297 thousands.b) For each increase of 1 thousand downloads, the expected revenue is estimated to increase by $ 3.7297 thousands.c) For each decrease of 1 thousand dollars in expected revenue, the expected number of downloads is estimated to increase by 3.7297 thousands.d) For each increase of 1 thousand dollars in expected revenue, the expected number of downloads is estimated to increase by 3.7297 thousands.

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