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Chapter 9: Problem 10

Determine which of the matrices are stochastic. \(\left[\begin{array}{rrr}.5 & .2 & .3 \\ .2 & .3 & .2 \\ .3 & .4 & .1 \\ 0 & 1 & 4\end{array}\right]\)

Short Answer

Expert verified
Based on the calculations: Row 1: \(0.5 + 0.2 + 0.3 = 1\) Row 2: \(0.2 + 0.3 + 0.2 = 0.7\) Row 3: \(0.3 + 0.4 + 0.1 = 0.8\) Row 4: \(0 + 1 + 4 = 5\) We can conclude that matrix A is not a stochastic matrix, since the sums of the elements in rows 2, 3, and 4 are not equal to 1.

Step by step solution

01

Identify and define the given matrix

Let's first identify the given matrix, which we will call matrix A. Matrix A is a 3x4 matrix given as follows: \(A = \left[\begin{array}{ccc} 0.5 & 0.2 & 0.3 \\ 0.2 & 0.3 & 0.2 \\ 0.3 & 0.4 & 0.1 \\ 0 & 1 & 4 \end{array}\right]\)
02

Sum the elements in each row

To check whether the matrix is stochastic or not, we need to add the elements in each row and see if the sum is equal to 1. Let's calculate the sum of elements in each row separately. Row 1: \(0.5 + 0.2 + 0.3 = 1\) Row 2: \(0.2 + 0.3 + 0.2 = 0.7\) Row 3: \(0.3 + 0.4 + 0.1 = 0.8\) Row 4: \(0 + 1 + 4 = 5\)
03

Check if the sums equal 1

Now that we have found the sums for each row, let's see if they equal 1: Row 1: Sum = 1 Row 2: Sum 鈮� 1 Row 3: Sum 鈮� 1 Row 4: Sum 鈮� 1 The sums of elements in rows 2, 3, and 4 do not equal 1.
04

Determine if the matrix is stochastic

Since the sums of the elements in rows 2, 3, and 4 are not equal to 1, we can conclude that matrix A is not a stochastic matrix.

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Key Concepts

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

finite mathematics
Finite mathematics is a branch of mathematics that deals with topics that are finite in nature, as opposed to those that require calculus and infinite processes. It encompasses a variety of subjects, which include set theory, matrix algebra, probability, and more.
Finite mathematics is primarily used in fields like business, economics, and the social sciences. It provides tools for modeling and solving real-world problems in these disciplines. For instance, you might use finite mathematics to model the behavior of a market or to predict the outcome of an election.
Understanding finite mathematics is crucial because it forms the foundation for more complex mathematical concepts that you may encounter later in your studies or professional work.
matrix operations
Matrix operations are procedures that can be performed on matrices, which are arrays of numbers arranged in rows and columns. These operations include addition, subtraction, multiplication, and more complex ones like finding the determinant or the inverse of a matrix.
Matrices are essential tools in many areas such as physics, computer science, engineering, and economics. One basic operation is matrix addition, which involves adding corresponding elements from two matrices of the same dimensions.
Another important operation is matrix multiplication, which is performed by multiplying rows by columns. This operation is not commutative, meaning that the order of multiplication affects the result. When dealing with matrices, always remember to check the dimensions to ensure you're performing operations correctly.
probability theory
Probability theory is a mathematical framework for quantifying uncertainty and making predictions about random events. It forms the backbone of many statistical methodologies used throughout science, engineering, finance, and everyday decision-making.
In probability theory, we frequently work with concepts like random variables, distributions, and stochastic processes. A matrix is called stochastic if each row sums to 1 and all elements are non-negative, representing transitions between states in a Markov chain.
Stochastic matrices are used in various applications such as modeling the probability of web pages being clicked on in a search engine or predicting weather patterns. Understanding probability theory and stochastic matrices thus allows for advanced modeling techniques that are crucial in both scientific research and practical applications.

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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. In the long run, what percentage of the commuters will be using public transportation?

At a certain university, three bookstores-the University Bookstore, the Campus Bookstore, and the Book Mart-currently serve the university community. From a survey conducted at the beginning of the fall quarter, it was found that the University Bookstore and the Campus Bookstore each had \(40 \%\) of the market, whereas the Book Mart had \(20 \%\) of the market. Each quarter the University Bookstore retains \(80 \%\) of its customers but loses \(10 \%\) to the Campus Bookstore and \(10 \%\) to the Book Mart. The Campus Bookstore retains \(75 \%\) of its customers but loses \(10 \%\) to the University Bookstore and \(15 \%\) to the Book Mart. The Book Mart retains \(90 \%\) of its customers but loses \(5 \%\) to the University Bookstore and \(5 \%\) to the Campus Bookstore. If these trends continue, what percentage of the market will each store have at the beginning of the second quarter? The third quarter?

Compute the steady-state matrix of each stochastic matrix. \(\left[\begin{array}{ccc}1 & .2 & .3 \\ 0 & .4 & .2 \\ 0 & .4 & .5\end{array}\right]\)

The transition matrix for a Markov process is given by \(T=\begin{array}{l}\text { State } 1 \\ \text { State } 2\end{array}\left[\begin{array}{ll}\frac{1}{2} & \frac{3}{4} \\ \frac{1}{2} & \frac{1}{4}\end{array}\right]\) and the initial-state distribution vector is given by $$ X_{0}=\begin{array}{l} \text { State } 1 \\ \text { State } 2 \end{array} \quad\left[\begin{array}{l} \frac{1}{3} \\ \frac{2}{3} \end{array}\right] $$ Find \(T X_{0}\) and interpret your result with the aid of a tree diagram.

Rewrite each absorbing stochastic matrix so that the absorbing states appear first, partition the resulting matrix, and identify the submatrices \(R\) and \(S\). \(\left[\begin{array}{rrrr}.4 & .2 & 0 & 0 \\ .2 & .3 & 0 & 0 \\ 0 & .3 & 1 & 0 \\ .4 & .2 & 0 & 1\end{array}\right]\)
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