Find the steady State vector for the transition matric 享
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![### Transition Matrices and Steady State Vectors
**Problem Statement:**
Find the steady state vector for the given transition matrix.
**Matrix Representation:**
\[
\begin{pmatrix}
\frac{3}{7} & \frac{4}{7} \\
\frac{2}{7} & \frac{5}{7}
\end{pmatrix}
\]
\[
\times
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
### Explanation of the Diagram
In this problem, we are given a transition matrix, which is a 2x2 matrix often used in Markov Chains to describe the probabilities of moving from one state to another. We are asked to find the steady-state vector, which is a vector that remains unchanged after the transition matrix has been applied to it.
- The left side of the equation represents the transition matrix:
\[
\begin{pmatrix}
\frac{3}{7} & \frac{4}{7} \\
\frac{2}{7} & \frac{5}{7}
\end{pmatrix}
\]
Each element inside this matrix can be interpreted as the transition probability from one state to another.
- The middle part represents the steady-state vector \(\mathbf{x}\) being multiplied by the transition matrix:
\[
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
- The right side shows that after applying the transition matrix to the steady-state vector, it remains the same:
\[
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
In other words, we need to solve for \(\mathbf{x}\) such that:
\[
\begin{pmatrix}
\frac{3}{7} & \frac{4}{7} \\
\frac{2}{7} & \frac{5}{7}
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
This can be set up as a system of linear equations and solved to find the steady-state vector.
### Steps to Solve:
1. Set up the system of linear equations from the matrix multiplication.
2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e80d594-14a8-4713-8ea5-1fda205a6b86%2F78a969c6-1fe9-426a-b053-e04d4100bcc7%2Fev8o67p.png&w=3840&q=75)
Transcribed Image Text:### Transition Matrices and Steady State Vectors
**Problem Statement:**
Find the steady state vector for the given transition matrix.
**Matrix Representation:**
\[
\begin{pmatrix}
\frac{3}{7} & \frac{4}{7} \\
\frac{2}{7} & \frac{5}{7}
\end{pmatrix}
\]
\[
\times
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
### Explanation of the Diagram
In this problem, we are given a transition matrix, which is a 2x2 matrix often used in Markov Chains to describe the probabilities of moving from one state to another. We are asked to find the steady-state vector, which is a vector that remains unchanged after the transition matrix has been applied to it.
- The left side of the equation represents the transition matrix:
\[
\begin{pmatrix}
\frac{3}{7} & \frac{4}{7} \\
\frac{2}{7} & \frac{5}{7}
\end{pmatrix}
\]
Each element inside this matrix can be interpreted as the transition probability from one state to another.
- The middle part represents the steady-state vector \(\mathbf{x}\) being multiplied by the transition matrix:
\[
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
- The right side shows that after applying the transition matrix to the steady-state vector, it remains the same:
\[
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
In other words, we need to solve for \(\mathbf{x}\) such that:
\[
\begin{pmatrix}
\frac{3}{7} & \frac{4}{7} \\
\frac{2}{7} & \frac{5}{7}
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
This can be set up as a system of linear equations and solved to find the steady-state vector.
### Steps to Solve:
1. Set up the system of linear equations from the matrix multiplication.
2
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