stochastic process problems


Question Description:

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Please help to solve this problem and show the solution in steps, urgently. We have more to be solved. Document Preview: Arrivals of passengers at a taxi stand form a Poisson process L with rate ?; passengers come singly and they wait patiently for their turn until a taxi shows up. Taxis arrive empty to the same stand according to Poisson process M with rate µ ; and each taxicab waits there until a ride shows up (if there were not only waiting passengers). Let Xt = Lt – Mt , t>=0, a) The process X = (X t) is a compound Poisson process that is, it has the form Xt = YNt where Y0 =0 and Yn = Z1 + Z2 +…+ Zn characterize the process N. Characterize the random variables Z1 , Z2 ,…; are they independent, what is their distribution? Is N independent of Y? b) Compute (enough to write down an explicit expression) P {Xt = 3}, P {Xt = -2}, P {Xt = 0}, Interpret what these probabilities are. c) Y = (Yn) n ? N is a Markov chain, what is its state space? Classify its states when ? > µ, and when ? < µ. Give intuitive justifications. Continuation. In the preceding problem, we now modify the taxicab behavior, when a taxicab arrives to find 3 taxicabs there (and therefore no passengers) it leaves immediately. So, the number Xt has to be in the set D = {-3, -2, -1, 0, 1, 2, …}. Show that X still has the form Xt = YNt but the Markov chain Y has transition probabilities different from those in the preceding problem. a) Compute the probabilities Pij = P{Y n+1 =j/Yn = i} i,j ? D b) Classify the states when ? < µ. Compute the limiting probabilities pj = limn-infinity P{Xn = j}. d) What can you say about limt-infinity Pi {Xt = j}. Attachments: zadachki-2rev....odt

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