Problem 25 - 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. Explain why the transition matrix does what we want it to do.

Step-by-Step Solution

Answer

The transition matrix shows how subscribers move between newspapers, ensuring the new distribution matches these movements each week.

Step 1: Understanding the Matrices

The state matrix \(X\) represents the distribution of Tribune and Picayune subscribers. Specifically, \(T\) is the number who get the Tribune, and \(P\) is the number who get the Picayune. The transition matrix \(Q\) defines how subscribers switch between the two newspapers each week.

Step 2: Analyze the Transition Matrix

The transition matrix \(Q\) is given as: \( Q = \begin{bmatrix} 0.90 & 0.20 \ 0.10 & 0.80 \end{bmatrix} \). This matrix shows probabilities: 0.90 of Tribune subscribers remain with Tribune, 0.10 switch to Picayune; 0.80 of Picayune subscribers remain with Picayune, 0.20 switch to Tribune. The columns indicate the destination, while the rows indicate the origin of the subscribers.

Step 3: Apply the Transition Matrix to Current State

By multiplying the current state matrix \(X\) by the transition matrix \(Q\), we calculate the distribution for the next week. Specifically, \(QX = \begin{bmatrix} 0.90T + 0.20P \ 0.10T + 0.80P \end{bmatrix} \). This multiplication distributes current subscribers according to the rates specified in the transition matrix.

Step 4: Verify Subscriber Movement

The first component of \(QX\) (\(0.90T + 0.20P\)) represents the new Tribune subscribers (those who stayed from Tribune plus those who switched from Picayune). The second component (\(0.10T + 0.80P\)) represents the new Picayune subscribers.

Key Concepts

State Matrix

To understand state matrices, think of them as snapshots that capture the status of different elements in a system, at a specific point in time. In our exercise, the state matrix \(X\) illustrates the number of subscribers to each newspaper—the Tribune (\(T\)) and the Picayune (\(P\))—during a given week.

• The state matrix is like a "current status report" for subscriber numbers.
• It enables us to keep track of subscribers at a particular moment.

In matrix notation, \(X\) is represented as \(\begin{pmatrix} T \ P \end{pmatrix}\). By analyzing this matrix, we get insights into how many individuals are engaged with each newspaper before any changes occur.

Probability

The concept of probability is essential for making predictions about future states. In our scenario, it describes the likelihood of subscribers either continuing with their current newspaper or switching to the other.

• Probability ranges between 0 and 1, where 0 signifies impossibility and 1 signifies certainty.
• These probabilities help determine the transitions between states—that is, how subscribers might shift from one newspaper to another.

For example, within the transition matrix, a 0.90 probability indicates that 90% of Tribune subscribers will stay subscribed to the Tribune next week. Meanwhile, a 0.10 probability means that 10% of them will move to the Picayune. Such probabilities are crucial for modeling predictable changes in subscriber behavior.

Subscriber Movement

Subscriber movement is the term used to describe the shifts between different states—in this case, between Tribune and Picayune subscriptions. It essentially captures how many people are expected to change or maintain their subscriptions in the given time frame.

• 0.90 of Tribune subscribers remain Tribune subscribers, indicating customer loyalty.
• 0.10 of Tribune subscribers switch to the Picayune, highlighting potential churn.
• 0.80 of Picayune subscribers remain, while 0.20 shift to Tribune.

These movements are not just random but are structured and predictable, thanks to the transition matrix. It reflects customer behavior patterns and helps businesses anticipate changes by aligning advertising or marketing strategies.

Matrix Multiplication

Matrix multiplication provides a systematic way to combine two matrices, producing a new matrix that aids in understanding transitions over time. Here, multiplying the transition matrix \(Q\) by the state matrix \(X\) allows us to predict the new state of newspaper subscriptions for the following week.

• The multiplication calculation is done by taking each row of the transition matrix and multiplying it by the column from the state matrix.
• Each element in the resulting matrix signifies potential new subscriber numbers for the corresponding newspaper.

Mathematically, this is expressed as:\[QX = \begin{pmatrix} 0.90T + 0.20P \ 0.10T + 0.80P \end{pmatrix}\]This formula helps in illustrating transitions and obtaining projections, thereby supporting strategic decisions in various fields, from marketing to resource planning.