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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- Exercise 48 If we throw a fair coin 10,000 times, what is the probability that it lands on “tail” at least 5,500 times? (i) Write down the exact probability in sum notation. (ii) Estimate the probability with the Chebyshev inequality. (iii) Estimate the probability with the Poisson‐Approximation. (iv) Estimate die probability with the normal distribution. Exercise 49 Let X ∼ NegBinom(900, 1/4 ). Estimate the probability P(X ≥ 3000) (i) with the Markov inequality, (ii) with the normal distribution. Hint: Use the fact that a negative binomial random variable can be written as a sum of geometric random variables. Exercise 50 (i) Let Xn ∼ Poisson(10), n ≥ 1, be independent random variables. Estimate the probability that P(9 ≤ S40 ≤ 12). (ii) Let Xn ∼ Exp(4), n ≥ 1, be independent random variables. Estimate the probability that P (1/100 * S100 ≥ 1 2).Suppose that a production process changes state according to a Markov chain on [25] state space S = {0, 1, 2, 3} whose transition probability matrix is given by a) Determine the limiting distribution for the process. b) Suppose that states 0 and 1 are “in-control,” while states 2 and 3 are deemed “out-of-control.” In the long run, what fraction of time is the process out-of-control?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.
- Find the limiting distribution for this Markov chain.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.A market analysis of car purchasing trends in a certain region has concluded that a family purchases a new car once every three years. They found that a small car is replaced with another small car 80% of the time and that a large car is replaced with another large car 60% of the time. (a) Define a transition matrix P for a Markov chain that describes the buying patterns. (Note that each “step” here is a 3-year period.) (b) Find the N-step transition matrix corresponding to 12 years from now. (c) If there are currently 40,000 small cars and 50,000 large cars in the region today, what is the prediction for the number of each in 12 years’ time? (d) Does the distribution of small/large cars stabilize?i. Find the eigenvalues of P.ii. Diagonalize P by finding an invertible matrix Q and diagonal matrix D such that P = QDQ−1 .iii. Use this to write P N and compute lim N→∞ P N . You’ll need the fact that limn→∞ (2/ 5)N = 0. This gives the stable proportions.
- Consider a Markov chain with transition matrix, find the stationary distribution.The random variables ξ, ξ1, ξ2, . . . are independent and identically distributed with distribution P (ξ = 0) = 1/4 and P (ξ = j) = c/j for j = 1, 2, Let X0 = 0 and Xn = max(ξ1, . . . , ξn) for n = 1, 2, . . .. What value must c take? Explain why {Xn, n = 0, 1, 2,..... } is a Markov Write down the transition Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). Calculate P (Xn = 0). Calculate P (X4 = 3, X2 = 1|X1 = 3).From yrs of teaching experience , an english teacher knows that her student's score will be va random variabel between variance = 25 adn mean = 75 how many students in general ,would have to take exam with a probability of atleast 0.9 , such that the class average would be within 5 of 75 dont use CLT , use markov /Chevyshev
- The computer center at Rockbottom University has been experiencing computer downtime. Let us assume that the trials of an associated Markov process are defined as one-hour periods and that the probability of the system being in a running state or a down state is based on the state of the system in the previous period. Historical data show the following transition probabilities. To From Running Down Running 0.90 0.10 Down 0.20 0.80 (a) If the system is initially running, what is the probability of the system being down in the next hour of operation? (b) What are the steady-state probabilities of the system being in the running state and in the down state? (Enter your probabilities as fractions.) Running?1= Down?2=DO NOT COPY OTHER SOLUTIONS. USE A TRANSITION PROBABILITY MATRIX WHEN SOLVING. Consider the following Markov chain. Assuming that the process starts in state 4, what is the probability that we eventually reach the recurrent class {1,2}?A fair die is tossed repeatedly. Let Xn be the number of 6’s obtained in the first n tosses. Show that {Xn : n = 1, 2,...} is a Markov chain. Then find its transition probability matrix, specify the classes and determine which are recurrent and which are transient.