Find Absorbing States in a Markov Chain Transition Matrix

Math Problem Statement

Find all absorbing states for this transition matrix. Is it the transition matrix for an absorbing Markov chain?

Start 3 By 3 Table 1st Row 1st Column 0.21 2nd Column 0.08 3rd Column 0.71 2nd Row 1st Column 0.39 2nd Column 0 3rd Column 0.61 3rd Row 1st Column 0 2nd Column 0 3rd Column 1 EndTable

0.21

0.08

0.71

0.39

0.61

Question content area bottom

Part 1

Find all absorbing states for the transition matrix. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

The absorbing state(s) is/are

enter your response here.

(Type a whole number. Use a comma to separate answers as needed.)

There are no absorbing states.

Solution

Identifying Absorbing States:

An absorbing state in a Markov chain is a state where, once entered, the process cannot leave. Mathematically, a state $i$ is absorbing if the $i$ -th row of the transition matrix has a 1 on the diagonal ( $P[i][i] = 1$ ) and 0 everywhere else.

Transition Matrix:

0.21 & 0.08 & 0.71 \\ 0.39 & 0 & 0.61 \\ 0 & 0 & 1 \end{bmatrix}$$ #### Step 1: Checking Diagonal Entries - State 1: $$P[1][1] = 0.21$$ (not equal to 1, so not absorbing) - State 2: $$P[2][2] = 0$$ (not equal to 1, so not absorbing) - State 3: $$P[3][3] = 1$$ (equal to 1, so it is absorbing) ### **Answer to Part 1:** - The absorbing state(s) is/are **3**. --- ### Part 2: Is this the transition matrix for an absorbing Markov chain? An **absorbing Markov chain** has at least one absorbing state, and it must be possible to reach an absorbing state from any other state (directly or indirectly). - **Step 1**: Identify absorbing states: $$\text{State 3 is absorbing.}$$ - **Step 2**: Check if all other states can eventually reach an absorbing state: - From State 1, there's a $$0.71$$ probability of transitioning to State 3. - From State 2, there's a $$0.61$$ probability of transitioning to State 3. Since all states can eventually reach an absorbing state, this is an **absorbing Markov chain**. --- ### Final Answer: 1. **Absorbing State**: $$\text{State 3.}$$ 2. **Yes, this is the transition matrix for an absorbing Markov chain.** --- Do you have further questions, or would you like a detailed explanation? Here are some related questions for deeper understanding: 1. What are the key properties of absorbing Markov chains? 2. How do you compute the expected number of steps to absorption? 3. What is the stationary distribution of this Markov chain? 4. Can you construct a Markov chain with multiple absorbing states? 5. How does one calculate absorption probabilities in an absorbing Markov chain? **Tip:** Always ensure that the sum of probabilities in each row of a transition matrix equals 1, as this is essential

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Math Problem Analysis

Mathematical Concepts

Markov Chains
Absorbing States
Transition Matrices

Formulas

Absorbing state condition: P[i][i] = 1, P[i][j] = 0 for all j ≠ i

Theorems

Definition of absorbing states in Markov chains
Properties of absorbing Markov chains

Suitable Grade Level

Grades 11-12 and Undergraduate

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