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 s_1 is a transient state.
Select an option
unanswered
State s_5 is a recurrent state.
Select an option
unanswered
State s_3 is a transient state.
Select an option
unanswered
There are two recurrent classes.(true or false)
Select an option
unanswered
State s_1 is a transient state.
False.
To determine whether each statement is true or false, we need to understand the concepts of transient and recurrent states in a Markov chain.
Recall that a transient state is a state in which there is a non-zero probability of eventually leaving and never returning. On the other hand, a recurrent state is a state in which there is a non-zero probability of returning to that state infinitely many times.
Now let's analyze each statement:
1. State s_1 is a transient state.
Looking at the transition probabilities, we can see that there is a probability of transitioning from s_1 to s_2 with a probability of 1. However, there is no probability of transitioning back from s_2 to s_1. This means that once we leave s_1, we can never return to it. Therefore, statement 1 is true.
2. State s_5 is a recurrent state.
Looking at the transition probabilities, we can see that there is a probability of transitioning from s_5 to s_4 with a probability of 1/2. Similarly, there is also a probability of transitioning back from s_4 to s_5 with a probability of 1/4. This means that there is a possibility of returning to state s_5 infinitely many times. Therefore, statement 2 is true.
3. State s_3 is a transient state.
Looking at the transition probabilities, we can see that there is a probability of transitioning from s_3 to s_2 with a probability of 1/10. However, there is no probability of transitioning back from s_2 to s_3. This means that once we leave s_3, we can never return to it. Therefore, statement 3 is true.
4. There are two recurrent classes.
Recall that a recurrent class is a set of states within which all states are recurrent to each other. In the given Markov chain, we can identify two recurrent classes: {s_2, s_3, s_4, s_5, s_6} and {s_7}. Each class has a set of states that can transition between themselves with non-zero probabilities. Therefore, statement 4 is true.
In summary:
1. True
2. True
3. True
4. True