A 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, 60% will remain listening to the news after the station break that occurs each half hour, while 40% will switch to the music station at the station break. Of the listeners who are tuned to the music station, 70% will switch to the news station at the station break, while 30% will remain listening to the music. Suppose everyone is listening to the news at 8:15 A.M. Complete parts a through c. a. Give the stochastic matrix that describes how the radio listeners tend to change stations at each station break, Label the rows and columns. Let N stand for "News" and M stand for "Music." Complete the stochastic matrix below. From: To: 0.60 0.70 0.40 0.30 (Type integers or decimals.) b. Give the initial state vector. Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice. (Type integers or decimals.) OA. Xo c. Xo = c. What percentage of listeners will be listening to the music station at 9:25 A.M. (after station breaks at 8:30 and 9:00 A.M.Y? At 9:25 A.M. % of listeners will be listening to the music station. (Type an integer.)
A 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, 60% will remain listening to the news after the station break that occurs each half hour, while 40% will switch to the music station at the station break. Of the listeners who are tuned to the music station, 70% will switch to the news station at the station break, while 30% will remain listening to the music. Suppose everyone is listening to the news at 8:15 A.M. Complete parts a through c. a. Give the stochastic matrix that describes how the radio listeners tend to change stations at each station break, Label the rows and columns. Let N stand for "News" and M stand for "Music." Complete the stochastic matrix below. From: To: 0.60 0.70 0.40 0.30 (Type integers or decimals.) b. Give the initial state vector. Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice. (Type integers or decimals.) OA. Xo c. Xo = c. What percentage of listeners will be listening to the music station at 9:25 A.M. (after station breaks at 8:30 and 9:00 A.M.Y? At 9:25 A.M. % of listeners will be listening to the music station. (Type an integer.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Educational Website Content: Stochastic Processes and Radio Listener Behavior**
---
**Example Problem: Radio Listener Dynamics in a Remote Village**
A remote village receives radio broadcasts from two stations: a news station and a music station. At each half-hour station break, listeners may switch stations. Of those tuned to the news station, 60% remain, while 40% switch to music. Conversely, of those tuned to the music station, 70% switch to news, while 30% continue with music.
**Task: Analyze Listener Transitions at 8:15 A.M.**
---
**a. Construct the Stochastic Matrix:**
Define transitions between stations as a stochastic matrix, with rows and columns labeled for "News" (N) and "Music" (M):
| From/To | N | M |
|---------|-----|----|
| N | 0.60| 0.40 |
| M | 0.70| 0.30 |
**b. Determine the Initial State Vector:**
Select the correct initial distribution of listeners. The vector represents the percentage of the audience initially tuned to each station.
Options:
- (i) x₀ = \[\begin{bmatrix}1\\0\end{bmatrix} \]
- (ii) x₀ = \[\begin{bmatrix}0\\1\end{bmatrix} \]
_Answer:_
- x₀ = \[\begin{bmatrix} 1\\ 0 \end{bmatrix} \] (All are initially listening to the news station.)
**c. Calculate Music Station Listener Percentage at 9:25 A.M.:**
Determine what percentage of listeners will be tuned to the music station at 9:25 A.M. after station breaks at 8:30 and 9:00 A.M.
_At 9:25 A.M., _____ % of listeners will be tuned to the music station._
---
**Real-World Application: Weather Transition Prediction**
Additional Problem Context: Consider analyzing weather progression. If today's weather is good:
- 50% chance for continued good weather tomorrow.
- 30% chance for indifferent weather.
- 20% chance for bad weather.
Predictions can be made using similar stochastic models.
---
Educators may use this case to help students understand stochastic processes, transition matrices, and real-life applications, such as demographics analysis and natural event predictions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2F0fd31b50-0601-41f6-b2ec-4d955b18d191%2F2lcekl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Content: Stochastic Processes and Radio Listener Behavior**
---
**Example Problem: Radio Listener Dynamics in a Remote Village**
A remote village receives radio broadcasts from two stations: a news station and a music station. At each half-hour station break, listeners may switch stations. Of those tuned to the news station, 60% remain, while 40% switch to music. Conversely, of those tuned to the music station, 70% switch to news, while 30% continue with music.
**Task: Analyze Listener Transitions at 8:15 A.M.**
---
**a. Construct the Stochastic Matrix:**
Define transitions between stations as a stochastic matrix, with rows and columns labeled for "News" (N) and "Music" (M):
| From/To | N | M |
|---------|-----|----|
| N | 0.60| 0.40 |
| M | 0.70| 0.30 |
**b. Determine the Initial State Vector:**
Select the correct initial distribution of listeners. The vector represents the percentage of the audience initially tuned to each station.
Options:
- (i) x₀ = \[\begin{bmatrix}1\\0\end{bmatrix} \]
- (ii) x₀ = \[\begin{bmatrix}0\\1\end{bmatrix} \]
_Answer:_
- x₀ = \[\begin{bmatrix} 1\\ 0 \end{bmatrix} \] (All are initially listening to the news station.)
**c. Calculate Music Station Listener Percentage at 9:25 A.M.:**
Determine what percentage of listeners will be tuned to the music station at 9:25 A.M. after station breaks at 8:30 and 9:00 A.M.
_At 9:25 A.M., _____ % of listeners will be tuned to the music station._
---
**Real-World Application: Weather Transition Prediction**
Additional Problem Context: Consider analyzing weather progression. If today's weather is good:
- 50% chance for continued good weather tomorrow.
- 30% chance for indifferent weather.
- 20% chance for bad weather.
Predictions can be made using similar stochastic models.
---
Educators may use this case to help students understand stochastic processes, transition matrices, and real-life applications, such as demographics analysis and natural event predictions.
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