Exercise: Recurrent and transient states
4 points possible (graded)
Consider the following transition probability graph:
This figure depicts a Markov chain with seven states. The possible states are S_1, S_2, up to S_7; S_1 through S_6 are depicted as a row of circles in the upper-middle part of the diagram and placed from left to right in this order, and S_7 is placed by itself directly below S_3. In this figure, transitions are depicted by arrows from a circle (representing the source) to a circle (representing the destination). The source and the destination may be the same, and transition arrows may be labelled with a number or a variable indicating the probability that if we start at the source, we will be at the given destination at the next step. Above the row of six circles are self-loops within the states S_2, S_3, through S_6. The probabilities of the self loops are as follows: 1/2 for S_2, 4/10 for S_3, 3/4 for S_4, 1/4 for S_5, and 1/2 for S_6. Between the circles representing the states S_1 through S_6 are transitions. The probability of the transition from S_1 to S_2 is 1 and from S_2 to S_1 is 1/2. There is no transition from S_2 to S_3, but the probability from S_3 to S_2 is 1/10. The transition from S_3 to S_4 has probability 2/10 and there's no transition from S_4 to S_3. The transition from S_4 to S_5 has probability 1/4 and from S_5 to S_4 is 1/2. Lastly, the probability from S_5 to S_6 is 1/4 and from S_6 to S_5 is 1/2. Below the row of circles, there are only two transition arrows: one from S_3 to S_7 with a probability of 3/10 and a self-loop within S_7 with a probability of 1.
Decide whether each of the following statements is true or false.
State is a transient state.
Select an option
unanswered
State is a recurrent state.
Select an option
unanswered
State is a transient state.
Select an option
unanswered
There are two recurrent classes.
Select an option
unanswered
1. False
2. True
3. True
4. False
State S1 is a transient state.
1. False
2. True
3. True
4. False
State S_7 is a transient state.
False. A transient state is a state that has a non-zero probability of eventually reaching an absorbing state, while state S_7 has a self-loop with a probability of 1, making it an absorbing state. Therefore, state S_7 is not a transient state.
To determine whether a state is transient or recurrent in a Markov chain, we need to analyze the probability of reaching that state from any other state.
- State S_1:
To determine if it is transient or recurrent, we need to analyze whether it is possible to return to S_1 after leaving it. In this case, we can see that there is a self-loop from S_2 to S_1 with a probability of 1/2, which means there is a possibility of returning to S_1. Therefore, the statement "State S_1 is a transient state" is false.
- State S_2:
Similarly, we can see that there is a self-loop from S_2 to itself with a probability of 1/2. This means that it is possible to return to S_2 after leaving it. Therefore, the statement "State S_2 is a transient state" is false.
- State S_3:
There is a transition arrow from S_3 to S_2 with a probability of 1/10, but there is also a self-loop within S_3 with a probability of 4/10. This means that there is a probability of leaving S_3 and returning to it as well. Therefore, the statement "State S_3 is a transient state" is false.
- State S_4:
There is a transition arrow from S_4 to S_5 with a probability of 1/4, but there is also a self-loop within S_4 with a probability of 3/4. This means that there is a probability of leaving S_4 and returning to it as well. Therefore, the statement "State S_4 is a transient state" is false.
- State S_5:
There is a transition arrow from S_5 to S_6 with a probability of 1/4, but there is also a self-loop within S_5 with a probability of 1/2. This means that there is a probability of leaving S_5 and returning to it as well. Therefore, the statement "State S_5 is a transient state" is false.
- State S_6:
There is a transition arrow from S_6 to S_5 with a probability of 1/2, but there is also a self-loop within S_6 with a probability of 1/2. This means that there is a probability of leaving S_6 and returning to it as well. Therefore, the statement "State S_6 is a transient state" is false.
- State S_7:
There is a self-loop within S_7 with a probability of 1. This means that there is a probability of leaving S_7 and returning to it as well. Therefore, the statement "State S_7 is a transient state" is false.
From the analysis above, we can conclude that none of the states (S_1 to S_7) are transient. Thus, the statement "There are two recurrent classes" is false.