Problem 26 - 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. If the conditions do not change from week to week, then \(Q\) remains the same and we have what's known as a Stochastic Process \({ }^{10}\) because Week \(n\) 's numbers are found by computing \(Q^{n} X .\) Choose a few values of \(n\) and, with the help of your classmates and calculator, find out how many people get each paper for that week. You should start to see a pattern as \(n \rightarrow \infty\).

Step-by-Step Solution

Answer

100 residents get the Tribune and 50 get the Picayune as \( n \rightarrow \infty \).

Step 1: Understand Initial State Matrix

We start by recognizing the state matrix \( X \) where \( T+P=150 \) because there are a total of 150 residents. Initially, we can set an arbitrary start, such as all residents receiving the Tribune, \( X = \begin{bmatrix} 150 \ 0 \end{bmatrix} \), or any other distribution.

Step 2: Apply Transition Matrix for One Week

Use the transition matrix \( Q \) to determine next week's distribution. Compute \( QX = \begin{bmatrix} 0.90 & 0.20 \ 0.10 & 0.80 \end{bmatrix} \begin{bmatrix} T \ P \end{bmatrix} \). If we start with \( T = 150 \) and \( P = 0 \), then the result will be \( \begin{bmatrix} 135 \ 15 \end{bmatrix} \), indicating 135 subscribe to Tribune and 15 to Picayune in week 1.

Step 3: Calculate for Subsequent Weeks (Iterate)

Repeat the multiplication to find \(X_2 = QQX\), \(X_3 = Q^3X\), and so on. For instance, for week 2, compute \( Q \begin{bmatrix} 135 \ 15 \end{bmatrix} \approx \begin{bmatrix} 123 \ 27 \end{bmatrix} \). Continue this process for additional weeks, recalculating the new matrix using \(Q\) from previous week's matrix.

Step 4: Identify the Pattern

Continue calculating for several weeks. You will observe that the numbers approach a steady state around \(X = \begin{bmatrix} 100 \ 50 \end{bmatrix} \), where it shows balance as both counts stabilize. This is called the steady-state distribution.

Step 5: Confirm with Calculations

Compute further, using week 5 or 6 values, like \( Q^6X \), to see that it consistently results in \( \begin{bmatrix} 100 \ 50 \end{bmatrix} \) regardless of initial distribution. This confirms that as \( n \rightarrow \infty \), \( T = 100 \) and \( P = 50 \) in this equilibrium.

Key Concepts

Transition Matrix

In the context of stochastic processes, a transition matrix plays a crucial role. It helps in determining the movement within a system from one state to another.
In our example of Pedimaxus, the transition matrix \( Q \) is structured to show how subscribers switch between the two newspapers over a week.

• Each row in the transition matrix corresponds to a state from which transitions occur.
• The columns represent the states to which transitions lead.
• The values in the matrix are probabilities that residents will move from one subscription to another.

Specifically, the first row \( \begin{bmatrix} 0.90 & 0.20 \end{bmatrix}\) indicates those initially subscribed to the Pedimaxus Tribune: there's a 90% chance they remain subscribed to it, and a 10% chance they switch to the Sasquatchia Picayune. The second row \( \begin{bmatrix} 0.10 & 0.80 \end{bmatrix}\) gives the probabilities for those currently with the Picayune.
Importantly, each row sums to 1, reflecting total probability conservation from one state to another. This provides a complete picture of subscription tendencies within the village.

State Matrix

The state matrix is a representation of the current or initial distribution of states in the system. For the village in question, it's essentially how many people subscribe to each newspaper at a given time.
We define it using matrix \( X = \begin{bmatrix} T \ P \end{bmatrix} \), where:

• \( T \) is the number of residents subscribing to the Pedimaxus Tribune.
• \( P \) is the number of residents subscribing to the Sasquatchia Picayune.

Initially, you might start with everyone subscribed to one paper, say the Tribune, making \( X = \begin{bmatrix} 150 \ 0 \end{bmatrix} \). Alternatively, start with a different distribution, but remember, the total must always equal 150 because that's the full population.
The accuracy in the initial setup of the state matrix is vital since it dictates the direction and pattern of subsequent calculations as every week progresses.

Steady-State Distribution

The steady-state distribution is when the state matrix reaches equilibrium, showing no more change despite further iterations of the process. This happens when the distribution stabilizes such that it no longer fluctuates each week.
In our Pedimaxus example, as you apply the transition matrix repeatedly, you'll notice the distribution of subscribers converging to \( \begin{bmatrix} 100 \ 50 \end{bmatrix} \).

• This outcome means 100 residents consistently subscribe to the Tribune.
• 50 residents remain with the Picayune.

Over time, even if you start with different initial distributions, as \( n \to \infty \), the result will stabilize at \( X = \begin{bmatrix} 100 \ 50 \end{bmatrix} \).
This shows a remarkable property of Markov processes, emphasizing that the initial state becomes less influential over time regarding the final equilibrium of a stochastic system.

Matrix Multiplication

Matrix multiplication is a fundamental operation you use to forecast how systems evolve over time in stochastic processes. In our scenario, it's how we use the transition matrix \( Q \) on the state matrix \( X \) to project future distributions.
Here are some key points:

• Multiply each row element of \( Q \) with the corresponding element of \( X \) and sum them to get the new entry for the resulting state matrix.
• For instance, \( QX \) returns a new state matrix showing week 2's subscription.

With matrix \( Q = \begin{bmatrix} 0.90 & 0.20 \ 0.10 & 0.80 \end{bmatrix} \) and \( X = \begin{bmatrix} 135 \ 15 \end{bmatrix} \) from week 1, multiplication results in \( \begin{bmatrix} 123 \ 27 \end{bmatrix} \) for week 2.
By continually applying the multiplication \( Q^n \times X \) for weeks \( n \), you simulate ongoing changes, illustrating the dynamic shifts within the system and arriving at the steady-state distribution. This operation is essential in predicting the evolution of Markov Chains.