Problem 4 Classify the states of a Markov ... [FREE SOLUTION]

Chapter 8: Problem 4

Classify the states of a Markov chain with transition matrix $$ p=\left(\begin{array}{llll} p & q & 0 & 0 \\ 0 & 0 & p & q \\ p & q & 0 & 0 \\ 0 & 0 & p & q \end{array}\right). $$ where $p+q=1, p \geq 0, q \geq 0$.

Short Answer

Expert verified

Classify states into two recurrent classes: $ \{1, 3\} $ and $ \{2, 4\} $.

Step by step solution

Understand the Structure of the Markov Chain

A Markov chain is represented by the transition matrix provided. The matrix describes the probabilities of transitioning from one state to another. Here, each element $ p_{ij} $ of the matrix corresponds to the probability of moving from state $ i $ to state $ j $.

Analyze the Transition Matrix

The transition matrix given is:\[ \begin{pmatrix} p & q & 0 & 0 \ 0 & 0 & p & q \ p & q & 0 & 0 \ 0 & 0 & p & q \end{pmatrix} \] Observe that states are paired (1 with 3, and 2 with 4) showing that 1 always leads to 2 or stays at 1, and similarly for the other pairs.

Identify Communicating Classes

Two states $ i $ and $ j $ are in the same communicating class if both can be reached from each other. Here, this forms two classes: $ \{1, 3\} $ and $ \{2, 4\} $. State 1 can reach state 3 and vice versa, similarly for states 2 and 4.

Analyze the Recurrence or Transience of Classes

A class is recurrent if, starting from any state in the class, any other state in that class will eventually be returned to almost surely. Since states 1 and 3 can only move to each other (and similarly 2 and 4), each class $ \{1, 3\} $ and $ \{2, 4\} $ is closed. This indicates both classes are recurrent.

Classify Each State

According to the analysis, states 1 and 3 form one recurrent communicating class, and states 2 and 4 form another recurrent communicating class. Therefore, none of the states are transient.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transition Matrix

In a Markov chain, the transition matrix is key. It provides a complete picture of how likely states are to change. Each element in the matrix denotes a specific transition probability. For example, in our problem:

Element $ p_{ij} $ is the probability of moving from state $ i $ to state $ j $.
The rows of the matrix sum up to 1, which reflects that a state must transition to some other state, possibly including itself.

The transition matrix in the exercise shows transition behavior between four states:\[\begin{pmatrix} p & q & 0 & 0 \0 & 0 & p & q \p & q & 0 & 0 \0 & 0 & p & q \end{pmatrix}\]The matrix has a distinctive pattern, forming pairs of states, namely (1, 3) and (2, 4), which indicates specific movement patterns between these states.

Communicating Classes

Communicating classes in a Markov chain categorize states based on reachability. Two states are in the same communicating class if each can be reached from the other. This is crucial:

It determines which states in the chain can influence each other through transitions.
A communicating class is the backbone of the Markov chain's structure.

In our exercise, the states form two such communicating classes:

Class $ \{1, 3\} $: States 1 and 3 reach each other through their transition probabilities.
Class $ \{2, 4\} $: Similarly, states 2 and 4 can reach one another.

These classes reveal that pairs of states interact within their sets but not across to others. Each class is isolated from the other in terms of transitions.

Recurrent States

Recurrent states are crucial in understanding long-term behavior in a Markov chain. A state is recurrent if, starting from this state, there is a certainty of returning to it. In our chain:

Each class $ \{1, 3\} $ and $ \{2, 4\} $ is closed, meaning all transitions occur within these pairs.
This closure leads to recurrence, as once you enter a class, you cannot exit.

Thus, states 1, 3, 2, and 4 are recurrent. Regardless of where you start in a class, you will eventually revisit states within it. Essentially, no state can be left permanently, and transitions continue endlessly within the communicating classes.

State Classification

Classifying states in a Markov chain connects their behavior over time. Each state fits into categories like recurrent or transient. Understanding this involves:

Identifying which states are recurrent, signaling perpetual return visits over time.
Transient states, which, in our exercise, surprisingly do not exist.

For our exercise matrix, states 1 and 3 belong to one recurrent class, and states 2 and 4 form another. Both classes are distinct and do not mix. This organization reveals that the system bounces back and forth between a limited set of states. State classification underlines the Markov chain's stability and predictability, ensuring all actions are within a defined, looping set over time.

91影视

Classify the states of a Markov chain with transition matrix $$ p=\left(\begin{array}{llll} p & q & 0 & 0 \\ 0 & 0 & p & q \\ p & q & 0 & 0 \\ 0 & 0 & p & q \end{array}\right). $$ where \(p+q=1, p \geq 0, q \geq 0\).

Short Answer

Step by step solution

Understand the Structure of the Markov Chain

Analyze the Transition Matrix

Identify Communicating Classes

Analyze the Recurrence or Transience of Classes

Classify Each State

Key Concepts

Transition Matrix

Communicating Classes

Recurrent States

State Classification

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