Discussion
Suppose \(f_{ij}\) denotes the probability of the first visit to \(j\) from \(i\) in a Markov chain with finite state space. The following are two statements :
(I) If C is a closed communicating class of persistent states, then for any transient state \(i\) which leads to \(j, k \in C\), \(f_{ij} = f_{ik}\).
(II) If C is a single closed communicating class and if \(i \to j\) where \(i \notin C\) and \(j \in C\), then \(f_{ij} = 1\).
Which of the following is true ?
Answer: B
Both (I) and (II) are true.
(I) Inside a closed communicating class C of persistent states, once the chain enters C it stays there and visits every state of C with probability 1. So starting from a transient \(i\), the chain reaches \(j \in C\) exactly when it enters C, and the same holds for \(k \in C\). Both \(f_{ij}\) and \(f_{ik}\) equal the probability of absorption into C.
(II) In a finite chain, the states outside the closed classes are transient and are left for good with probability 1. If C is the only closed class, the chain must eventually enter C, and then it hits every state of C. So \(f_{ij} = 1\) for \(i \notin C\), \(j \in C\).
Finiteness matters for (II): with infinitely many transient states the chain could wander off and never reach C. As stated, both hold, so the other options are wrong.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development