Question

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

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

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 \(P = \left( {\begin{array}{*{20}{c}}{.7}&{.6}\\{.3}&{.4}\end{array}} \right)\).

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 \({{\mathop{\rm x}\nolimits} _0} = \left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right)\).

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.