Let {X„} be a time homogeneous Markov Chain with sample space {1,2, 3, 4} and transition matrix P = Does this Markov Chain converge to a stationary distribution? If it does, find the stationary distribution. If not, explain. O -131 3
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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.)Let 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.Let Zm represent the outcome during the nth roll of a fair dice. Define the Markov chain X, to be the maximum outcome obtained so far after the nth roll, i.e., X, = max {Z1, Z2,..., Zn}. What is the transition probability p22 of the Markov chain {Xn}?
- Which of the following best describes the long-run probabilities of a Markov chain {Xn: n = 0, 1, 2, ...}? O the probabilities of eventually returning to a state having previously been in that state O the fraction of time the states are repeated on the next step O the fraction of the time being in the various states in the long run O the probabilities of starting in the various states • Previous Not saved SimpfunGive two interpretation of what the first entry of the distribution (the limiting distribution of the markov chain) tells you based on the definition of a limiting distribution. Your answer should be written for a non-mathematician and should consist of between 1 and 3 complete sentences without mathematical symbols or terminology. Both interpretations should be written in this way.(Exponential Distribution) must be about Markov Chain. The time between supernova explosions in the known universe is exponentially distributed with an average of 1 month. According to Mr. Spock, the spaceship Atlantic's scientist, a supernova explosion may deform the structure of the Higgs Boson, and may cause the extinction of the entire universe, with a probability of 1 in 118 billion. Assuming that the age of the universe is about 16 billion years, calculate the probability that such an event will cause the universe to disappear in the next 10 billion years.
- Suppose 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 Poisson random variable with mean λ = 20. Estimate the probability P(X ≥25) based on: (a) Markov inequality. (b) Chebyshev inequality. (c) Chernoff bound. (d) Central limit theorem.Find the limiting distribution for this Markov chain. Then give an interpretation of what the first entry of the distribution you found tells you based on the definition of a limiting distribution. Your answer should be written for a non-mathematician and should consist of between 1 and 3 complete sentences without mathematical symbols or terminology.
- Suppose I flip n independent biased coins such that the jth coin has probability j/n of being heads, for j = 1,...,n. Let Hn be a random variable equal to the total number of heads. (a) What is the expectation of H,? (b) What is the variance of Hn? (c) Use Markov's inequality to derive an upper bound on P(Hn 2 9n/10).Consider a Markov chain {Xn}n≥0 having the following transition diagram: For this chain, there are two recurrent classes R1 = {6, 7} and R2 = {1, 2, 5}, and one transient class R3 = {3, 4}. Find the period of state Find f33 and f22. Starting at state 3, find the probability that the chain is absorbed into R1. 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.Explain how you can tell this Markov chain has a limiting distribution and how you could compute it. Your answer should refer to the relevant Theorems.