For each one of the following definitions of the state Xk at time k (for k=1,2,…), determine whether the Markov property is satisfied by the sequence X1,X2,….

Question

A fair six-sided die (with sides labelled 1,2,…,6) is rolled repeatedly and independently.

(a) Let Xk denote the largest number obtained in the first k rolls. Does the sequence X1,X2,… satisfy the Markov property?

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(b) Let Xk denote the number of 6's obtained in the first k rolls, up to a maximum of ten. (That is, if ten or more 6's are obtained in the first k rolls, then Xk=10.) Does the sequence X1,X2,… satisfy the Markov property?

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(c) Let Yk denote the result of the kth roll. Let X1=Y1, and for k≥2, let Xk=Yk+Yk−1. Does the sequence X1,X2,… satisfy the Markov property?

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(d) Let Yk=1 if the kth roll results in an odd number; and Yk=0 otherwise. Let X1=Y1, and for k≥2, let Xk=Yk⋅Xk−1. Does the sequence X1,X2,… satisfy the Markov property?

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Let Yk be the state of some Markov chain at time k (i.e., it is known that the sequence Y1,Y2,… satisfies the Markov property).

(a) For a fixed integer r>0, let Xk=Yr+k. Does the sequence X1,X2,… satisfy the Markov property?

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(b) Let Xk=Y2k. Does the sequence X1,X2,… satisfy the Markov property?

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(c) Let Xk=(Yk,Yk+1). Does the sequence X1,X2,… satisfy the Markov property?

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