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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=(.7.6.3.4).
  2. The initial state vector is x0=(10).
  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 โ€œNewsโ€ and M represent โ€œMusicโ€.

The following table represents the listenerโ€™s behavior.

From:

To:

N

M

.7

.6

N

.3

.4

M

Therefore, the stochastic matrix is P=(.7.6.3.4).

02

Determine the initial state vector

b)

If P is a nร—n regular stochastic matrix,then according to Theorem 18,it has a unique steady-state vector q.Further, if x0 is any initial state and xk+1=Pxk for k=0,1,2,...., then the Markov chain {xk} converges to qas kโ†’โˆž.

The initial state vector is x0=(10) because 100% of the listeners are listening to the news at 8:15 A.M.

Thus, the initial state vector is x0=(10).

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 x2 as shown below:

x1=Px0=(.7.6.3.4)(10)=(0.7+00.3+0)=(0.70.3)

x2=Px1=(.7.6.3.4)(.7.3)=(0.49+0.180.21+0.12)=(0.670.33)

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.

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