Consider a Markov chain {X0,X1,…}, specified by the following transition probability graph.
P(X2=2∣X0=1)= - unanswered
Find the steady-state probabilities ð1, ð2, and ð3 associated with states 1, 2, and 3, respectively.
ð1= - unanswered
ð2= - unanswered
ð3= - unanswered
For n=1,2,…, let Yn=Xn−Xn−1. Thus, Yn=1 indicates that the nth transition was to the right, Yn=0 indicates that it was a self-transition, and Yn=−1 indicates that it was a transition to the left.
limn→∞P(Yn=1)= - unanswered
Is the sequence Y1,Y2,… a Markov chain?
- unanswered
Given that the nth transition was a transition to the right (Yn=1), find (approximately) the probability that the state at time n−1 was state 1 (i.e., Xn−1=1). Assume that n is large.
- unanswered
Suppose that X0=1. Let T be the first positive time index n at which the state is equal to 1.
E[T]= - unanswered
Does the sequence X1,X2,X3,… converge in probability to a constant?
- unanswered
Let Zn=max{X1,…,Xn}. Does the sequence Z1,Z2,Z3,… converge in probability to a constant?
- unanswered
4) No
7) No
8) Yes
1. 0.44
2. 1/9
2/9
6/9
3. 1/9
5. 2/5
E[T]=9
You guys are amazing, I love you
Somebody knows the answer to Q6...E[T]
To find the transition probability P(X2=2|X0=1), we need to look at the transition probability graph provided. In this case, we are looking for the probability of transitioning from state 1 to state 2 in two steps.
When given a transition probability graph, we need to identify the probabilities associated with each transition. In this case, we don't have the specific probabilities for the transitions, so we can't determine P(X2=2|X0=1) without additional information.
For finding the steady-state probabilities ð1, ð2, and ð3 associated with states 1, 2, and 3, respectively, we need to solve the system of equations:
ð1 * transition probability from state 1 to 1 + ð2 * transition probability from state 2 to 1 + ð3 * transition probability from state 3 to 1 = ð1
ð1 * transition probability from state 1 to 2 + ð2 * transition probability from state 2 to 2 + ð3 * transition probability from state 3 to 2 = ð2
ð1 * transition probability from state 1 to 3 + ð2 * transition probability from state 2 to 3 + ð3 * transition probability from state 3 to 3 = ð3
These equations represent the balance equations for a Markov chain. Solving this system of equations will give us the steady-state probabilities ð1, ð2, and ð3.
To find the limit as n approaches infinity of P(Yn=1), we need to analyze the behavior of Yn, which represents the transitions in the Markov chain. Since Yn takes values of 1, 0, and -1, we can consider the probabilities of transitions to the right, self-transitions, and transitions to the left.
If the Markov chain is ergodic, meaning it is irreducible and aperiodic, the limit as n approaches infinity of P(Yn=1) is the fraction of time the Markov chain spends making transitions to the right. However, if the Markov chain is not ergodic, this limit may not exist or might be different.
To determine whether the sequence Y1, Y2, ... is a Markov chain, we need to check for the Markov property. The Markov property states that the conditional probability distribution of the future states only depends on the present state and not on the past states. In this case, if the transitions in the Markov chain satisfy the Markov property, then the sequence Y1, Y2, ... is a Markov chain.
To find the probability that the state at time n-1 was state 1 (i.e., Xn-1=1) given that the nth transition was a transition to the right (Yn=1), we can use the concept of conditional probability. We need to calculate P(Xn-1=1|Yn=1).
If we assume that n is large, we can use the steady-state probabilities obtained earlier to approximate P(Xn-1=1|Yn=1).
To find E[T], the expected value of T (the first positive time index n at which the state is equal to 1), we need to consider the probability distribution of T. Specifically, we need to determine the probability that T takes each possible value and then calculate the weighted average of these values.
If the sequence X1, X2, X3, ... converges in probability to a constant, it means that as the sequence progresses, the probability of being in any particular state approaches a fixed value. To determine if this is the case, we need to analyze the behavior of the sequence and see if it exhibits such convergence.
Finally, to determine if the sequence Z1, Z2, Z3, ... converges in probability to a constant, we need to analyze the behavior of the sequence of maximum states reached. If the maximum state reached keeps changing or does not approach a fixed value, then the sequence does not converge in probability to a constant. Otherwise, it does converge.