[.6 Let P = be a transition matrix. Which one of the following vectors .4 .7 10 4 10 is the steady-state vector for this transition matrix? Justify your response by demonstrating it is the steady- state vector by a computation that verifies the definition of steady-state vector.
[.6 Let P = be a transition matrix. Which one of the following vectors .4 .7 10 4 10 is the steady-state vector for this transition matrix? Justify your response by demonstrating it is the steady- state vector by a computation that verifies the definition of steady-state vector.
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![**Title: Identifying the Steady-State Vector for a Transition Matrix**
Let \( P = \begin{bmatrix}
0.6 & 0.3 \\
0.4 & 0.7
\end{bmatrix} \) be a transition matrix. Which one of the following vectors
\[ q = \begin{bmatrix}
\frac{3}{7} \\
\frac{4}{7}
\end{bmatrix} \]
\[ q = \begin{bmatrix}
\frac{3}{4} \\
\frac{1}{4}
\end{bmatrix} \]
\[ q = \begin{bmatrix}
\frac{3}{4} \\
1
\end{bmatrix} \]
\[ q = \begin{bmatrix}
\frac{6}{10} \\
\frac{4}{10}
\end{bmatrix} \]
is the steady-state vector for this transition matrix? Justify your response by demonstrating it is the steady-state vector by a computation that verifies the definition of a steady-state vector.
**Instructions for Verification:**
To verify which vector \( q \) is the steady-state vector, perform the following computation:
1. Multiply the transition matrix \( P \) by each vector \( q \).
2. Check if \( Pq = q \). If the equation holds true, then \( q \) is the steady-state vector for the transition matrix \( P \).
**Definition of Steady-State Vector:**
A vector \( q \) is a steady-state vector for a transition matrix \( P \) if \( Pq = q \). This means that applying the transition matrix to the vector doesn’t change the vector, indicating that it has reached a stable state.
**Note:**
The steady-state vector, also known as the stationary distribution, represents a state where probabilities remain constant upon subsequent applications of the transition probabilities represented by the matrix \( P \).
This exercise demonstrates an important concept in Markov Chains and the convergence behavior of systems described by transition matrices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00a48d79-f805-418c-a679-1c91878d1d75%2F8d00c764-70bd-41c7-a294-f38b6a84aeba%2F5l52o2e_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Identifying the Steady-State Vector for a Transition Matrix**
Let \( P = \begin{bmatrix}
0.6 & 0.3 \\
0.4 & 0.7
\end{bmatrix} \) be a transition matrix. Which one of the following vectors
\[ q = \begin{bmatrix}
\frac{3}{7} \\
\frac{4}{7}
\end{bmatrix} \]
\[ q = \begin{bmatrix}
\frac{3}{4} \\
\frac{1}{4}
\end{bmatrix} \]
\[ q = \begin{bmatrix}
\frac{3}{4} \\
1
\end{bmatrix} \]
\[ q = \begin{bmatrix}
\frac{6}{10} \\
\frac{4}{10}
\end{bmatrix} \]
is the steady-state vector for this transition matrix? Justify your response by demonstrating it is the steady-state vector by a computation that verifies the definition of a steady-state vector.
**Instructions for Verification:**
To verify which vector \( q \) is the steady-state vector, perform the following computation:
1. Multiply the transition matrix \( P \) by each vector \( q \).
2. Check if \( Pq = q \). If the equation holds true, then \( q \) is the steady-state vector for the transition matrix \( P \).
**Definition of Steady-State Vector:**
A vector \( q \) is a steady-state vector for a transition matrix \( P \) if \( Pq = q \). This means that applying the transition matrix to the vector doesn’t change the vector, indicating that it has reached a stable state.
**Note:**
The steady-state vector, also known as the stationary distribution, represents a state where probabilities remain constant upon subsequent applications of the transition probabilities represented by the matrix \( P \).
This exercise demonstrates an important concept in Markov Chains and the convergence behavior of systems described by transition matrices.
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