Problem 29 - Precalculus

Question

Consider the following scenario. In the small village of Pedimaxus in the country of Sasquatchia, all 150 residents get one of the two local newspapers. Market research has shown that in any given week, \(90 \%\) of those who subscribe to the Pedimaxus Tribune want to keep getting it, but \(10 \%\) want to switch to the Sasquatchia Picayune. Of those who receive the Picayune, \(80 \%\) want to continue with it and \(20 \%\) want switch to the Tribune. We can express this situation using matrices. Specifically, let \(X\) be the 'state matrix' given by $$ X=\left[\begin{array}{l} T \\ P \end{array}\right] $$ where \(T\) is the number of people who get the Tribune and \(P\) is the number of people who get the Picayune in a given week. Let \(Q\) be the 'transition matrix' given by $$ Q=\left[\begin{array}{ll} 0.90 & 0.20 \\ 0.10 & 0.80 \end{array}\right] $$ such that \(Q X\) will be the state matrix for the next week. Show that \(S Y=X_{s}\) for any matrix \(Y\) of the form $$ Y=\left[\begin{array}{r} y \\ 150-y \end{array}\right] $$ This means that no matter how the distribution starts in Pedimaxus, if \(Q\) is applied often enough, we always end up with 100 people getting the Tribune and 50 people getting the Picayune.

Step-by-Step Solution

Answer

After repeated transitions, 100 people get the Tribune, and 50 get the Picayune.

Step 1: Understanding the Scenario

We have a village with 150 residents, each subscribing to either the Pedimaxus Tribune or the Sasquatchia Picayune. The transition rates between the two newspapers are expressed in matrix form. The transition matrix \( Q \) tells us how people switch subscriptions each week.

Step 2: Defining the Matrices

Define the initial state matrix \( X \) as \( \begin{bmatrix} T \ P \end{bmatrix} \) where \( T \) and \( P \) are the number of Tribune and Picayune subscribers, respectively. Define the transition matrix \( Q \) as \( \begin{bmatrix} 0.90 & 0.20 \ 0.10 & 0.80 \end{bmatrix} \). This reflects the percentage of subscribers from each newspaper who choose to stay or switch.

Step 3: Applying the Transition Matrix

Multiply the state matrix \( X \) by the transition matrix \( Q \) to find the new distribution in the following week. We calculate: \[ QX = \begin{bmatrix} 0.90 & 0.20 \ 0.10 & 0.80 \end{bmatrix} \begin{bmatrix} T \ P \end{bmatrix} = \begin{bmatrix} 0.90T + 0.20P \ 0.10T + 0.80P \end{bmatrix} \]

Step 4: Solving for Steady State

The steady state distribution does not change when the transition matrix is applied. This means we have the equation: \[ \begin{bmatrix} T \ P \end{bmatrix} = \begin{bmatrix} 0.90T + 0.20P \ 0.10T + 0.80P \end{bmatrix} \]Solving these equations, we get two simultaneous equations:1. \( T = 0.90T + 0.20P \)2. \( P = 0.10T + 0.80P \)

Step 5: Solving Equations

To solve these equations, simplify each:1. \( T - 0.90T = 0.20P \) \( 0.10T = 0.20P \) \( T = 2P \)2. \( P - 0.80P = 0.10T \) \( 0.20P = 0.10T \) \( 2P = T \)Both conditions lead to \( T = 2P \), meaning twice as many people have the Tribune as have the Picayune.

Step 6: Solving for Total Subscribers

Given \( T + P = 150 \) and \( T = 2P \), substitute to find\( 2P + P = 150 \)\( 3P = 150 \)\( P = 50 \)Then \( T = 2P = 100 \).

Step 7: Conclusion - Steady State Verification

The conclusion from this steady state analysis shows that after many applications of \( Q \), the distribution will be 100 subscribers for the Tribune and 50 for the Picayune, irrespective of the initial distribution. This is verified by the nature of the matrices defined in the problem.

Key Concepts

Transition Matrix

In the fascinating world of Markov Chains, a transition matrix is a powerful tool. It visualizes how a system changes over time. In our example, it represents how village residents switch from subscribing to one newspaper to another. The matrix, denoted as \( Q \), captures the probabilities of these transitions in a given period. Each entry in the transition matrix corresponds to the likelihood that a person will continue subscribing to their current newspaper or switch to the other.

• The element \( Q_{11} = 0.90 \) indicates that 90% of the Pedimaxus Tribune's subscribers choose to continue with it.
• Similarly, \( Q_{22} = 0.80 \) suggests that 80% of the Sasquatchia Picayune's subscribers stay with their newspaper.
• The element \( Q_{12} = 0.20 \) shows that 20% of Picayune readers switch to the Tribune.
• Finally, \( Q_{21} = 0.10 \) stands for the 10% of Tribune subscribers who move to the Picayune.

This matrix structure ensures that each row sums to 1, reinforcing the concept that each person's choice is represented through this probability model. The transition matrix provides a snapshot of change, showing how populations evolve over time.

State Matrix

The state matrix, denoted as \( X \), captures the current distribution of subscribers between different states—in this case, newspapers. It is a simple column matrix that reflects the tally of people associated with each state (or newspaper).For instance:

• \( T \) stands for the number of people subscribing to the Pedimaxus Tribune.
• \( P \) denotes those who receive the Sasquatchia Picayune.

Mathematically, the state matrix is expressed as: \[X = \begin{bmatrix} T \ P \end{bmatrix}\] This setup becomes essential when multiplied by the transition matrix, as it predicts changes in the distribution of states over a period. Each multiplication with \( Q \) brings forth a new state matrix, illustrating the dynamic behavior of subscribers in the village.The interaction between the state and transition matrices offers a powerful method to forecast the state matrix for upcoming weeks. This dynamic outlook helps us understand how the number of subscribers to each newspaper will adapt and progress over time.

Steady State

One critical concept in studying Markov Chains is the steady state. A steady state occurs when the system reaches a distribution that remains unchanged by further applications of the transition matrix.In simpler terms, it's the point at which the numbers of newspaper subscribers no longer fluctuate with each subsequent week. In our scenario, the steady state is realized when the ongoing application of the transition matrix results in fixed numbers of Tribune and Picayune readers.Mathematically, the steady state is achieved when: \[\begin{bmatrix} T \ P \end{bmatrix} = \begin{bmatrix} 0.90T + 0.20P \ 0.10T + 0.80P \end{bmatrix}\]The challenge is to find values for \( T \) and \( P \) that satisfy these equations, leading us to:

• \( T = 2P \)
• where \( T + P = 150 \)

These conditions mean twice as many readers subscribe to the Tribune compared to the Picayune. Ultimately, calculations reveal that the steady state consists of 100 Tribune and 50 Picayune subscribers. Understanding this balance helps highlight how systems evolve toward stability over time.