/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Compute the steady-state matrix ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 24

Compute the steady-state matrix of each stochastic matrix. \(\left[\begin{array}{ccccc}1 & 0 & \frac{1}{4} & \frac{1}{3} & 0 \\ 0 & 1 & 0 & \frac{1}{3} & \frac{1}{2} \\ 0 & 0 & \frac{1}{4} & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)

Short Answer

Expert verified
There are two possible steady-state matrices for the given stochastic matrix: \( S_1 = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ \end{array}\right] \) and \( S_2 = \left[\begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \end{array}\right] \)

Step by step solution

01

Find the steady-state vector s

The first step is to find the steady-state vector s, which satisfies the equation s * M = s, where M is the given stochastic matrix. Let the steady-state vector s = \(\left[ s_1, s_2, s_3, s_4, s_5 \right]\). We need to solve the system of linear equations: \(s_1 + s_3/4 + s_4/3 = s_1\) \(s_2 + s_4/3 + s_5/2 = s_2\) \(s_3/4 + s_4/3 = s_3\) \(s_3/2 + s_5/2 = s_4\) s_5 = 0 Also, the sum of the elements of the vector s must be equal to 1 (since it represents a probability distribution): \(s_1 + s_2 + s_3 + s_4 + s_5 = 1\)
02

Solve the system of linear equations

We can simplify the equations from Step 1 as follows: \(s_3/4 = 0\) \(s_4/3 = 0\) \(s_3/4 + s_4/3 = s_3\) \(s_3/2 = s_4\) s_5 = 0 \(s_1 + s_2 + s_3 + s_4 = 1\) Now we solve this system of linear equations. Since \(s_3/4 = 0\), we have that s_3 = 0. Similarly, from the second equation, we get s_4 = 0. Now, from the fourth equation, we have that \(s_3/2=s_4\), but both s_3 and s_4 are 0, so the equation holds. Lastly, we substitute s_3 = s_4 = s_5 = 0 in the last equation: \( s_1 + s_2 = 1 \) Since s_1 and s_2 represent probabilities, they must be non-negative. Therefore, we have two possibilities: (1) s_1 = 1 and s_2 = 0, and (2) s_1 = 0 and s_2 = 1.
03

Construct the steady-state matrix S

For each possibility of the steady-state vector, we will construct a steady-state matrix by repeating this vector in each row. For possibility (1): \(s_1 = 1\) and \(s_2 = 0\), we have the steady-state vector: \( s = \left[ 1, 0, 0, 0, 0 \right] \) and the steady-state matrix: \( S = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ \end{array}\right] \) For possibility (2): \(s_1 = 0\) and \(s_2 = 1\), we have the steady-state vector: \( s = \left[ 0, 1, 0, 0, 0 \right] \) and the steady-state matrix: \( S = \left[\begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \end{array}\right] \) There are two possible steady-state matrices for the given stochastic matrix: \( S_1 = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ \end{array}\right] \) and \( S_2 = \left[\begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \end{array}\right] \)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stochastic Matrix
A stochastic matrix is a special type of matrix used in probability theory. It is employed to describe transitions in a Markov chain, where each element represents the probability of moving from one state to another. The key property of a stochastic matrix is that each of its rows must add up to one, ensuring that the total probability of transitioning from a given state is 100%.
  • Row-stochastic Matrix: In this form, each row sums to one. Every element in the matrix corresponds to the probability of transitioning from one state to either a different state or staying in the same state.
  • Column-stochastic Matrix: Each column sums to one, and it denotes the probability distribution of states' outputs.
In the exercise, we work with a row-stochastic matrix.
Linear Equations
Linear equations are mathematical expressions that equate a linear combination of variables to a constant. They are crucial in various fields, including mathematics, engineering, and economics.
When we deal with stochastic matrices in our exercise, we solve a set of linear equations to find the steady-state vector. This vector must satisfy a key condition \(s \times M = s\), where \(s\) is the steady-state vector and \(M\) is the stochastic matrix. This means the vector remains unchanged after the matrix transformation.
  • Characteristics: Each equation represents a balance of probabilities, ensuring probabilities are conserved across states.
  • Solving Method: Techniques such as substitution or Gaussian elimination can be used to solve these linear equations efficiently.
The satisfying solutions of these equations help us find the steady-state matrix, which describes the equilibrium states.
Probability Distribution
Probability distribution represents how probabilities are spread over possible outcomes or values. It is a fundamental concept in statistics and probability theory. For stochastic processes, a probability distribution describes the likelihood of different outcomes over states or events.
In the context of stochastic matrices, the steady-state vector is a probability distribution because it contains the probabilities of being in each state once the system has stabilized.
  • Discrete Probability Distribution: Deals with discrete random variables, representing distinct results.
  • Continuous Probability Distribution: Involves continuous random variables with ranges of values.
In our exercise, the condition \(s_1 + s_2 + s_3 + s_4 + s_5 = 1\) ensures that the vector forms a valid probability distribution, as it sums up to one.
Steady-State Vector
The steady-state vector is a critical component in understanding long-term behavior in stochastic processes, such as Markov chains. It denotes a fixed vector of probabilities that remain constant after repeated applications of the stochastic matrix.
To find the steady-state vector, we solve \(s \times M = s\), where \(s\) is the vector and \(M\) is the stochastic matrix. This equilibrium condition implies that the application of the matrix does not change the probability distribution represented by the vector.
  • Unique Steady-State: Depending on the stochastic matrix, there may be one or multiple steady-state vectors.
  • Interpretation: It provides insight into stable states within systems where the probabilities of states no longer change.
In the exercise's solution, we identified two possible steady-state vectors, leading to two distinct steady-state matrices, each demonstrating possible outcomes of the system鈥檚 dynamics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Within a large metropolitan area, \(20 \%\) of the commuters currently use the public transportation system, whereas the remaining \(80 \%\) commute via automobile. The city has recently revitalized and expanded its public transportation system. It is expected that 6 mo from now \(30 \%\) of those who are now commuting to work via automobile will switch to public transportation, and \(70 \%\) will continue to commute via automobile. At the same time, it is expected that \(20 \%\) of those now using public transportation will commute via automobile and \(80 \%\) will continue to use public transportation. a. Construct the transition matrix for the Markov chain that describes the change in the mode of transportation used by these commuters. b. Find the initial distribution vector for this Markov chain. c. What percentage of the commuters are expected to use public transportation \(6 \mathrm{mo}\) from now?

Find \(X_{2}\) (the probability distribution of the system after two observations) for the distribution vector \(X_{0}\) and the transition matrix \(T\). \(X_{0}=\left[\begin{array}{l}.25 \\ .40 \\ .35\end{array}\right], T=\left[\begin{array}{ccc}.1 & .1 & .3 \\ .8 & .7 & .2 \\ .1 & .2 & .5\end{array}\right]\)

The payoff matrix for a game is $$ \left[\begin{array}{rrr} -3 & 3 & 2 \\ -3 & 1 & 1 \\ 1 & -2 & 1 \end{array}\right] $$ a. Find the expected payoff to the row player if the row player \(R\) uses the maximin pure strategy and the column player \(C\) uses the minimax pure strategy. b. Find the expected payoff to the row player if \(R\) uses the maximin strategy \(50 \%\) of the time and chooses each of the other two rows \(25 \%\) of the time, while \(C\) uses the minimax strategy \(60 \%\) of the time and chooses each of the other columns \(20 \%\) of the time. c. Which of these pairs of strategies is most advantageous to the row player?

Find the steady-state vector for the transition matrix. $$ \left[\begin{array}{ll} .9 & 1 \\ .1 & 0 \end{array}\right] $$

Determine whether the two-person, zero-sum matrix game is strictly determined. If a game is strictly determined, a. Find the saddle point(s) of the game. b. Find the optimal strategy for each player. c. Find the value of the game. d. Determine whether the game favors one player over the other. $$ \left[\begin{array}{rrr} -1 & 2 & 4 \\ 2 & 3 & 5 \\ 0 & 1 & -3 \\ -2 & 4 & -2 \end{array}\right] $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.