Let (X,} be a time homogeneous Markov Chain with sample space {1,2,3, 4}. Gi the transition matrix P = 1-131130 11311300113 11301 3
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Find the 2-step transition matrix.
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- 12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction randomly choose which way to go. Figure 3.28 (a) Construct the transition matrix for the Markov chain that models this situation. (b) Suppose we start with 15 robots at each junction. Find the steady state distribution of robots. (Assume that it takes each robot the same amount of time to travel between two adjacent junctions.)Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.The transition matrix of a Markov chain is given by 0 0 P = (a) Find two distinct stationary distributions of this Markov chain. (b) Find the general form of the stationary distribution. (c) If 70) = (, 4, ¿, 1, ) is the initial probability vector at time 0, ,π(m) = (금' 등,을 ,). 4'4'6' 6' 6 then show that limn 12 HIN O -160 2/3 HIN O 230116 O 0 -16 0 116
- Find the vector of stable probabilities for the Markov chain whose transition matrix isSuppose the transition matrix for a Markov Chain is T = stable population, i.e. an x0₂ such that Tx = x. ساله داد Find a non-zeroLet X be a random variable with sample space {1,2, 3} and probability distribu- (G 1 ). Find a transition matrix P such that the Markov chain {X„} tion T = simulates X.
- In certain parts s of the world, tuberculosis (TB) is present in 25% of the population. Chest X-ray used as a diagnostic tool has an accuracy of 99% detection of TB if present. Only in 0.5% of the cases, normal people get a positive diagnosis. Draw a transition chart to demonstrate the model. Obtain the transition matrix. What is the probability that a randomly chosen person tests negative? If a randomly chosen person tests positive, what is the probability that the person was disease free? Q.5 a. b. C. d.How many DAGS are Markov equivalent to the “chain DAG" X1 → X2 → · .. → X, (excluding itself)?A continuous time Markov chain on state space {1,2}1,2 has generator matrix Q_{11} = -1, Q_{12} = 1, Q_{21}=4, Q_{22} = -4Q11=-1,Q12=1,Q21=4,Q22 =-4. What is the stationary probability of state 1? a). 1/5 b). None of the choices. c). 4/5 d). 1/2 е). 1/4
