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BHUSHAN MISAL

BHUSHAN M.

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Questions asked

ANSWERED

Rachel Gore verified

Numerade educator

Let Xn be a Markov Chain with state space S = 0, 1, 2, · · ·. For each of the following transition probabilities,state if the chain is positive recurrent,null recurrent,or transient. If it is positive recurrent, give the stationary probability distribution : (a) Pi0 = 1 i+3 , Pi(i+1) = i+2 i+3 (∀i ∈ S) (b) Pi0 = i+2 i+3 , Pi(i+1) = 1 i+3 (∀i ∈ S) (c) Pi0 = 1 i 2+3 , Pi(i+1) = i 2+2 i 2+3 (∀i ∈ S) (d) Assume that we keep rolling on a fair dice. What is the expected number of rolls before we see k consecutive 5s.

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INSTANT ANSWER

(a) Given a rate λ Poisson process N(t) and an independent exponential (α) random variable X, what is the distribution of number of arrivals N(X) in the interval [0, X]? (b) Events occur according to a Poisson process with rate λ. Each time an event occurs, we must decide whether or not to stop, with our objective being to stop at the last event to occur prior to some specified time T, where T > 1 λ . That is, if an event occurs at time t, 0 ≤ t ≤ T, and we decide to stop, then we win if there are no additional events by time T, and we lose otherwise. If we do not stop when an event 1 occurs and no additional events occur by time T, then we lose. Also, if no events occur by time T, then we lose. Consider the strategy that stops at the first event to occur after some fixed time s, 0 ≤ s ≤ T. Compute the probability of winning when using the preceding strategy, for an optimally chosen s.

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ANSWERED

Rachel Gore verified

Numerade educator

Let Xn be a Markov Chain on state space {1,2,3,4,5} with transition matrix P = [0 0.5 0.5 0 0 0 0 0 0.2 0.8 0 0 0 0.4 0.6 1 0 0 0 0 0.5 0 0 0 0.5] (a) Is the chain irreducible? Is it periodic? (b) Suppose that X0 = 1. What is the expected number of steps until the pattern of states (2,5,1,3,4,1,2,5,1,3,4,1,2,5) appears? (c) Suppose that X0 = 1. What is the expected number of times the Markov Chain will see pattern of states 3412 before seeing pattern of states 251? (d) Again suppose that X0 = 1. What is the probability that the chain reaches the sequence 251 at n=10 the first time?

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