Homework Help is Here – Start Your Trial Now!
Math
Probability
Find the steady-state vector for the transition matrix. 0.6 0.4 0.1 0.4 0.2 0.4 0 0.4 0.5 X =

Find the steady-state vector for the transition matrix. 0.6 0.4 0.1 0.4 0.2 0.4 0 0.4 0.5 X =

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
See similar textbooks
icon
Related questions
Question
**Title: Finding the Steady-State Vector for a Transition Matrix** **Introduction:** In this exercise, we will find the steady-state vector for a given transition matrix. A steady-state vector is a probability vector that remains unchanged after successive applications of a transition matrix. This concept is widely used in Markov chains and various applications, such as predicting long-term behavior in stochastic processes. **Problem Statement:** Find the steady-state vector for the given transition matrix: \[ \begin{bmatrix} 0.6 & 0.4 & 0.1 \\ 0.4 & 0.2 & 0.4 \\ 0 & 0.4 & 0.5 \end{bmatrix} \] **Solution Approach:** 1. **Understanding the Transition Matrix:** - Each row of the matrix represents the transition probabilities from one state to all possible states. - The columns signify the probability of transitioning to that state from all other states. 2. **Finding the Steady-State Vector (\(X\)):** - We are tasked with finding a vector \(X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\) such that when the transition matrix is multiplied by \(X\), it results in the same vector \(X\). - Mathematically, it is represented as \(AX = X\), where \(A\) is the transition matrix. 3. **Steps to Find \(X\):** - Solve the equation \( (A - I)X = 0 \), where \(I\) is the identity matrix. - Ensure that the sum of the elements in \(X\) equals 1, as \(X\) is a probability vector. By solving these equations, we can find the steady-state vector which represents the long-term behavior of the system represented by the transition matrix. **Conclusion:** Understanding how to find and interpret a steady-state vector is crucial for analyzing and predicting the behavior of systems modeled by transition matrices. It provides insights into the probability distribution of states over the long term.
Transcribed Image Text:**Title: Finding the Steady-State Vector for a Transition Matrix** **Introduction:** In this exercise, we will find the steady-state vector for a given transition matrix. A steady-state vector is a probability vector that remains unchanged after successive applications of a transition matrix. This concept is widely used in Markov chains and various applications, such as predicting long-term behavior in stochastic processes. **Problem Statement:** Find the steady-state vector for the given transition matrix: \[ \begin{bmatrix} 0.6 & 0.4 & 0.1 \\ 0.4 & 0.2 & 0.4 \\ 0 & 0.4 & 0.5 \end{bmatrix} \] **Solution Approach:** 1. **Understanding the Transition Matrix:** - Each row of the matrix represents the transition probabilities from one state to all possible states. - The columns signify the probability of transitioning to that state from all other states. 2. **Finding the Steady-State Vector (\(X\)):** - We are tasked with finding a vector \(X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\) such that when the transition matrix is multiplied by \(X\), it results in the same vector \(X\). - Mathematically, it is represented as \(AX = X\), where \(A\) is the transition matrix. 3. **Steps to Find \(X\):** - Solve the equation \( (A - I)X = 0 \), where \(I\) is the identity matrix. - Ensure that the sum of the elements in \(X\) equals 1, as \(X\) is a probability vector. By solving these equations, we can find the steady-state vector which represents the long-term behavior of the system represented by the transition matrix. **Conclusion:** Understanding how to find and interpret a steady-state vector is crucial for analyzing and predicting the behavior of systems modeled by transition matrices. It provides insights into the probability distribution of states over the long term.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer