Find the steady-steee vector for the trausition matrix
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![## Finding the Steady-State Vector for a Transition Matrix
To determine the steady-state vector, we need to analyze the given transition matrix. The transition matrix is expressed as follows:
\[
\begin{pmatrix}
0.1 & 0.4 & 0.3 \\
0.1 & 0.4 & 0.3 \\
0.8 & 0.2 & 0.4
\end{pmatrix}
\]
We are tasked with finding the steady-state vector \( \mathbf{x} \) which is expressed as:
\[
\mathbf{x} = \begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
\]
In mathematical terms, the steady-state vector satisfies the equation:
\[
P \mathbf{x} = \mathbf{x}
\]
where \( P \) is the transition matrix. This equation can be rewritten as:
\[
(P - I)\mathbf{x} = \mathbf{0}
\]
where \( I \) is the identity matrix, and \( \mathbf{0} \) is the zero vector.
To solve for \( \mathbf{x} \), one would typically convert this problem into a system of linear equations.
- Diagram Explanation:
The image consists of the transition matrix and the steady-state vector written side by side. The transition matrix, enclosed in large parentheses, contains three rows with three columns of probabilities, representing state transitions. The steady-state vector \( \mathbf{x} \), also enclosed in large parentheses, consists of three components stacked vertically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e80d594-14a8-4713-8ea5-1fda205a6b86%2Fa53408fe-a2ac-4af2-85d3-83f491c3a5d3%2Fxus779.png&w=3840&q=75)
Transcribed Image Text:## Finding the Steady-State Vector for a Transition Matrix
To determine the steady-state vector, we need to analyze the given transition matrix. The transition matrix is expressed as follows:
\[
\begin{pmatrix}
0.1 & 0.4 & 0.3 \\
0.1 & 0.4 & 0.3 \\
0.8 & 0.2 & 0.4
\end{pmatrix}
\]
We are tasked with finding the steady-state vector \( \mathbf{x} \) which is expressed as:
\[
\mathbf{x} = \begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
\]
In mathematical terms, the steady-state vector satisfies the equation:
\[
P \mathbf{x} = \mathbf{x}
\]
where \( P \) is the transition matrix. This equation can be rewritten as:
\[
(P - I)\mathbf{x} = \mathbf{0}
\]
where \( I \) is the identity matrix, and \( \mathbf{0} \) is the zero vector.
To solve for \( \mathbf{x} \), one would typically convert this problem into a system of linear equations.
- Diagram Explanation:
The image consists of the transition matrix and the steady-state vector written side by side. The transition matrix, enclosed in large parentheses, contains three rows with three columns of probabilities, representing state transitions. The steady-state vector \( \mathbf{x} \), also enclosed in large parentheses, consists of three components stacked vertically.
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