Question

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.


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unanswered
State is a transient state.


Select an option
unanswered
There are two recurrent classes.


Select an option
unanswered

Answer 1

State S1 is a transient state.

Answer 2

1. False
2. True
3. True
4. False

Answer 3

1. False
2. True
3. True
4. False