Which of thefollowing processes Xn are Markovchains? (a) Theprocess Xn movesfrom

Question

Which of thefollowing processes Xn are Markovchains?

(a) Theprocess Xn movesfrom i to2i withprobability 1/2 and stays in i with probability 1/2.

(b) Theprocess Xn movesfrom i toi+ 1 with probability 1/2 and toi+ n with probability 1/2.

(c) Wheneverthe process Xn first arrives at astate i, it stays at that state for 5 steps andthen moves to state i +1.

(d) Wheneverthe process Xn first arrives at astate i, it stays at that state for a random amountof steps Ni where Ni has thePoisson distribution with parameter a, and thenmoves to state i +1; it is assumed thatthe Ni areindependent.

(e) IfXn1 = i then Xn = i withprobability 1/2 and Xn = i + 1 with probability 1/2; however, if the process has already stayedin i longer than 3steps then it moves to i +1.

(f) Theprocess Xn movesfrom i to2i withprobability 1/n andstays in i withprobability n-1/n.

(g)Xn = An whereAis a random variablewhere A = 1 with probability 1/2 and A = +1 withprobability 1/2.

Explanation / Answer

A Markov chain is one in which future states depend only on thepresent state (not any past states). a) Yes b) Yes c) No d) No e) No f) Yes g) Yes (this is a random walk)

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