Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Question 1: A small remote village receives radio broadcasts from two radio stations, a news station, and a music station. Of the listeners who are tuned to the news station, \(70\% \) will remain listening to the news after the station break that occurs each half-hour, while \(30\% \) will switch to the music station at the station break. Of the listeners who are tuned to the music station, \(60\% \) will switch to the news station at the station break, while \(40\% \) will remain listening to the music. Suppose everyone is listening to the news at 8:15 A.M.

a. Give the stochastic matrix that describes how the radio listeners tend to change stations at each station break. Label the rows and columns.

b. Give the initial state vector.

c. What percentage of the listeners will be listening to the music station at 9:25 A.M. (After the station breaks at 8:30 and 9:00 A.M.)?

Short Answer

Expert verified
  1. The stochastic matrix is \(P = \left( {\begin{array}{*{20}{c}}{.7}&{.6}\\{.3}&{.4}\end{array}} \right)\).
  2. The initial state vector is \({{\mathop{\rm x}\nolimits} _0} = \left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right)\).
  3. 33% of the listeners will be listening to the music station at 9:25 A.M.

Step by step solution

01

Determine the stochastic matrix

a)

Let \(N\) represent “News” and \(M\) represent “Music”.

The following table represents the listener’s behavior.

From:

To:

N

M

.7

.6

N

.3

.4

M

Therefore, the stochastic matrix is \(P = \left( {\begin{array}{*{20}{c}}{.7}&{.6}\\{.3}&{.4}\end{array}} \right)\).

02

Determine the initial state vector

b)

If \(P\) is a \(n \times n\) regular stochastic matrix,then according to Theorem 18,it has a unique steady-state vector \({\mathop{\rm q}\nolimits} \).Further, if \({{\mathop{\rm x}\nolimits} _0}\) is any initial state and \({{\mathop{\rm x}\nolimits} _{k + 1}} = P{{\mathop{\rm x}\nolimits} _k}\) for \(k = 0,1,2,....\), then the Markov chain \(\left\{ {{{\mathop{\rm x}\nolimits} _k}} \right\}\) converges to \({\mathop{\rm q}\nolimits} \)as \(k \to \infty \).

The initial state vector is \({{\mathop{\rm x}\nolimits} _0} = \left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right)\) because 100% of the listeners are listening to the news at 8:15 A.M.

Thus, the initial state vector is \({{\mathop{\rm x}\nolimits} _0} = \left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right)\).

03

Determine the percentage of the listeners who will be listening at 9.25 A.M.

c)

Two breaks occur between 8:15 A.M. and 9:25 A.M.

Compute \({{\mathop{\rm x}\nolimits} _2}\) as shown below:

\(\begin{array}{c}{{\mathop{\rm x}\nolimits} _1} = P{{\mathop{\rm x}\nolimits} _0}\\ = \left( {\begin{array}{*{20}{c}}{.7}&{.6}\\{.3}&{.4}\end{array}} \right)\left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.7 + 0}\\{0.3 + 0}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.7}\\{0.3}\end{array}} \right)\end{array}\)

\(\begin{array}{c}{{\mathop{\rm x}\nolimits} _2} = P{{\mathop{\rm x}\nolimits} _1}\\ = \left( {\begin{array}{*{20}{c}}{.7}&{.6}\\{.3}&{.4}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{.7}\\{.3}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.49 + 0.18}\\{0.21 + 0.12}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.67}\\{0.33}\end{array}} \right)\end{array}\)

At 9:25 A.M., 33% of the listeners will be listening to the news.

Thus, 33% of the listeners will be listening to the music station at 9:25 A.M.

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 Vaia!

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

In Exercises 27-30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation.

a. What is the dimension of range of T if T is one-to-one mapping? Explain.

b. What is the dimension of the kernel of T (see section 4.2) if T maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\)? Explain.

[M] Let \(A = \left[ {\begin{array}{*{20}{c}}7&{ - 9}&{ - 4}&5&3&{ - 3}&{ - 7}\\{ - 4}&6&7&{ - 2}&{ - 6}&{ - 5}&5\\5&{ - 7}&{ - 6}&5&{ - 6}&2&8\\{ - 3}&5&8&{ - 1}&{ - 7}&{ - 4}&8\\6&{ - 8}&{ - 5}&4&4&9&3\end{array}} \right]\).

  1. Construct matrices \(C\) and \(N\) whose columns are bases for \({\mathop{\rm Col}\nolimits} A\) and \({\mathop{\rm Nul}\nolimits} A\), respectively, and construct a matrix \(R\) whose rows form a basis for Row\(A\).
  2. Construct a matrix \(M\) whose columns form a basis for \({\mathop{\rm Nul}\nolimits} {A^T}\), form the matrices \(S = \left[ {\begin{array}{*{20}{c}}{{R^T}}&N\end{array}} \right]\) and \(T = \left[ {\begin{array}{*{20}{c}}C&M\end{array}} \right]\), and explain why \(S\) and \(T\) should be square. Verify that both \(S\) and \(T\) are invertible.

In Exercise 18, Ais an \(m \times n\) matrix. Mark each statement True or False. Justify each answer.

18. a. If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.

b. Row operations preserve the linear dependence relations among the rows of A.

c. The dimension of the null space of A is the number of columns of A that are not pivot columns.

d. The row space of \({A^T}\) is the same as the column space of A.

e. If A and B are row equivalent, then their row spaces are the same.

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

22. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 1}&{ - 13}&{ - 12.2}&{ - 1.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

See all solutions

Recommended explanations on Math Textbooks

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.

Sign-up for free