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?
- unanswered
(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?
- unanswered
(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?
- unanswered
(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?
- unanswered
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?
- unanswered
(b) Let Xk=Y2k. Does the sequence X1,X2,… satisfy the Markov property?
- unanswered
(c) Let Xk=(Yk,Yk+1). Does the sequence X1,X2,… satisfy the Markov property?
- unanswered
(a) Let Xk denote the largest number obtained in the first k rolls. Does the sequence X1,X2,… satisfy the Markov property?
Well, you see, the Markov property is all about forgetting the past and just focusing on the present. But in this case, the largest number obtained in the first k rolls depends on the previous rolls. So, it seems like this sequence is holding onto its past, like a hoarder who just can't let go. So, no, the Markov property is not satisfied here. Maybe it needs some therapy.
(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?
Ah, the number of 6's, the holy grail of dice rolling. Well, in this case, the sequence is only concerned with the current roll and the number of 6's obtained so far. It doesn't care about the individual rolls before. So, it's the epitome of living in the present moment, embracing the almighty Markov property. Congrats, my friend!
(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?
Huh, this one is interesting. Xk is the sum of the kth roll and the (k-1)th roll. So, in order to determine the value of Xk, you need to know the values of both Yk and Yk-1. It's like a dynamic duo that relies on each other, like Batman and Robin. But unfortunately, this means that the Markov property is not satisfied here. Looks like we won't be seeing any cape-wearing heroes in this sequence.
(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?
Ah, the odd and even rolls, a tale as old as time. In this case, the value of Xk is determined by the current roll and the previous value of Xk-1. It's like a never-ending cycle of odd and even, like a dance party that never stops. So, unfortunately, the Markov property isn't satisfied here. Looks like this party is going to have to find a new DJ.
Now, let's move on to the next set of definitions!
(a) For a fixed integer r>0, let Xk=Yr+k. Does the sequence X1,X2,… satisfy the Markov property?
Well, my friend, in this case, Xk is determined by a fixed value r and some future rolls. It's like having a crystal ball that predicts the future, but only if you know the value of r. So, unfortunately, the Markov property is not satisfied here. It's like trying to solve a puzzle without all the pieces. Good luck with that!
(b) Let Xk=Y2k. Does the sequence X1,X2,… satisfy the Markov property?
Oh, intriguing! This time, Xk is determined by squaring the value of Yk. It's like a mathematical love story, where Yk is the protagonist and Xk is the outcome of their epic romance. However, in order to determine Xk, you need the value of Yk itself. So, sorry to break it to you, but the Markov property is not satisfied here. Looks like true love doesn't always conquer all, especially when it comes to Markov chains.
(c) Let Xk=(Yk,Yk+1). Does the sequence X1,X2,… satisfy the Markov property?
Well, my friend, in this case, Xk is a tuple that includes the value of Yk and Yk+1. It's like a dynamic duo that always sticks together, like peanut butter and jelly. However, in order to determine the value of Xk, you need to know the values of both Yk and Yk+1, which means the Markov property is not satisfied here. It's like trying to separate peanut butter from jelly. It's just not gonna happen.
To determine whether the sequences satisfy the Markov property, we need to verify if the future state depends only on the current state and not on the past states.
(a) Let Xk denote the largest number obtained in the first k rolls. The sequence X1, X2, ... does not satisfy the Markov property because the future maximum number obtained is influenced by the entire history of the rolls, not just the current state.
(b) Let Xk denote the number of 6's obtained in the first k rolls, up to a maximum of ten. The sequence X1, X2, ... satisfies the Markov property because the future number of 6's obtained only depends on the current state (the number of 6's obtained) and not on the past rolls.
(c) Let Yk denote the result of the kth roll and Xk=Yk+Yk−1. The sequence X1, X2, ... satisfies the Markov property because the future state only depends on the current state (the sum of the current and previous rolls) and not on the past rolls.
(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. The sequence X1, X2, ... satisfies the Markov property because the future state only depends on the current state (whether the current roll is odd or even) and not on the past rolls.
For the given sequences:
(a) No, it does not satisfy the Markov property.
(b) Yes, it satisfies the Markov property.
(c) Yes, it satisfies the Markov property.
(d) Yes, it satisfies the Markov property.
Let's now move on to the second set of questions.
(a) For a fixed integer r>0, let Xk=Yr+k. The sequence X1, X2, ... satisfies the Markov property because the future state only depends on the current state (Yr+k) and not on the past states.
(b) Let Xk=Y2k. The sequence X1, X2, ... satisfies the Markov property because the future state only depends on the current state (Y2k) and not on the past states.
(c) Let Xk=(Yk,Yk+1). The sequence X1, X2, ... satisfies the Markov property because the future state only depends on the current state (the pair (Yk,Yk+1)) and not on the past states.
I hope this helps! Let me know if you have any further questions.
To determine whether a given sequence satisfies the Markov property, we need to check if the future behavior of the sequence depends only on the present state and not the past states.
(a) In this case, Xk denotes the largest number obtained in the first k rolls of a fair six-sided die. The Markov property is satisfied if the future behavior only depends on the current state Xk and not the past states. Here, the future behavior is determined solely by the next roll of the die, therefore, the Markov property is satisfied.
(b) Here, Xk denotes the number of 6's obtained in the first k rolls, up to a maximum of ten. The Markov property is satisfied if the future behavior only depends on the current state Xk and not the past states. However, in this case, the value of Xk depends not only on the current roll but also on the past rolls. Therefore, the Markov property is not satisfied.
(c) This time, Yk denotes the result of the kth roll of the die, and Xk is defined as Xk = Yk + Yk-1 for k >= 2. To check if the Markov property is satisfied, we need to see if the future behavior only depends on the current state Xk and not the past states. In this case, Xk is determined by the current roll and the previous roll, meaning that knowledge of Xk-1 is necessary to determine the future behavior. Therefore, the Markov property is not satisfied in this case.
(d) Here, Yk takes the value 1 if the kth roll results in an odd number, and 0 otherwise. Xk is defined as Xk = Yk * Xk-1 for k >= 2. Similarly, we need to check if the future behavior only depends on the current state Xk. In this case, Xk is determined by the current roll Yk and the past state Xk-1, meaning that knowledge of Xk-1 is necessary to determine the future behavior. Therefore, the Markov property is not satisfied in this case.
To answer the remaining questions, please provide the state transitions or equations that define the Markov chain.