Problem 3: Checking the Markov property
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,….
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?
Yes - correct
(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?
Yes - correct
(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?
No - correct
(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?
Yes - correct
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?
Yes - correct
(b) Let Xk=Y2k. Does the sequence X1,X2,… satisfy the Markov property?
Yes - correct
(c) Let Xk=(Yk,Yk+1). Does the sequence X1,X2,… satisfy the Markov property?
Yes - correct
"answered in full"
answers to "Problem 1: Steady-state convergence" .
Belonging to this very same problem set....
1.a. False
1.b. False
2.a. False
2.b. False
3.a. True
3.b. True
1-
a - yes
b- yes
c - no
d- yes
2-
yes for all
I have no doubt in my mind that those answers are 100% correct. Unless they're not, in which case, I have absolutely no idea what I'm talking about. But hey, at least I'm honest, right?
The Markov property states that the conditional probability of the future state only depends on the current state and is independent of the past states.
(a) For the sequence X1,X2,... where Xk denotes the largest number obtained in the first k rolls of a fair six-sided die, the Markov property is satisfied. The largest number obtained in the previous rolls does not affect the future rolls.
(b) For the sequence X1,X2,... where Xk denotes the number of 6's obtained in the first k rolls, up to a maximum of ten, the Markov property is satisfied. The number of 6's obtained in the previous rolls does not affect the future rolls.
(c) For the sequence X1,X2,... where Xk=Yk+Yk−1, the Markov property is not satisfied. The current state Xk depends on both the current state Yk and the previous state Yk-1, which means it depends on the past states and not just the current state.
(d) For the sequence X1,X2,... where Xk=Yk⋅Xk−1, the Markov property is satisfied. The current state Xk depends only on the current state Yk and the previous state Xk-1, not on the past states.
For the second set of questions:
(a) For the sequence X1,X2,... where Xk=Yr+k, the Markov property is satisfied. The future state Xk only depends on the current state Yr+k and is independent of the past states.
(b) For the sequence X1,X2,... where Xk=Y2k, the Markov property is satisfied. The future state Xk only depends on the current state Y2k and is independent of the past states.
(c) For the sequence X1,X2,... where Xk=(Yk,Yk+1), the Markov property is satisfied. The future state Xk only depends on the current state (Yk,Yk+1) and is independent of the past states.
To determine whether a sequence satisfies the Markov property, we need to check if the conditional probability distribution of the future state depends only on the current state and not on the past states.
In each case, we will consider the sequence X1, X2, ... and check if the Markov property is satisfied.
(a) For the sequence Xk representing the largest number obtained in the first k rolls of a fair six-sided die, the Markov property is satisfied. This is because the probability distribution of the largest number in the next roll only depends on the current largest number. Knowing the past rolls does not provide any additional information.
(b) For the sequence Xk representing the number of 6's obtained in the first k rolls (up to a maximum of ten), the Markov property is satisfied. The probability distribution of the next number of 6's only depends on the current number of 6's. The past rolls do not affect this probability.
(c) For the sequence Xk = Yk + Yk-1, where Yk represents the result of the kth roll, the Markov property is not satisfied. In this case, the value of Xk depends on both the current and previous rolls. Knowing the entire history of rolls is necessary to determine the future values of Xk.
(d) For the sequence Xk = Yk * Xk-1, where Yk represents 1 if the kth roll results in an odd number and 0 otherwise, the Markov property is satisfied. The probability distribution of the next value of Xk only depends on the current value of Xk-1. The past rolls are not needed to determine the future values of Xk.
For the next set of questions, we consider the sequence X1, X2, ... obtained from a given Markov chain sequence Y1, Y2, ...
(a) For Xk = Yr+k, where r is a fixed integer, the sequence X1, X2, ... satisfies the Markov property. The probability distribution of the future states only depends on the current state Xr+k and does not require knowledge of the past states.
(b) For Xk = Y2k, the sequence X1, X2, ... satisfies the Markov property. The probability distribution of X2k only depends on Y2k and does not require knowledge of the past states.
(c) For Xk = (Yk, Yk+1), the sequence X1, X2, ... satisfies the Markov property. The probability distribution of (Yk, Yk+1) only depends on the current pair of states and does not require knowledge of the past states.
In summary, the sequences X1, X2, ... satisfy the Markov property in cases (a), (b), (d), (a), (b), and (c). The Markov property is not satisfied in case (c).