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Chapter 3: Problem 6

Let \(P=\left[\begin{array}{ccc}\frac{1}{2} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{3} & \frac{2}{3} \\ \frac{1}{2} & \frac{1}{3} & 0\end{array}\right]\) be the transition matrix for a Markov chain with three states. Let \(\mathbf{x}_{0}=\left[\begin{array}{r}120 \\ 180 \\ 90\end{array}\right]\) be the initial state vector for the population. What proportion of the state 1 population will be in state 1 after two steps?.

Short Answer

Expert verified
The proportion of state 1 population in state 1 after two steps is 1.29 or 129%.

Step by step solution

01

Understanding Matrix Multiplication

To find the state distribution after two steps, we perform matrix multiplication. We know that the state vector after one step is given by multiplying the transition matrix \( P \) with the initial state vector \( \mathbf{x}_0 \). The formula is \( \mathbf{x}_1 = P \cdot \mathbf{x}_0 \).
02

Calculate First Step State Vector

Compute the first-step state vector \( \mathbf{x}_1 \) as follows: \[\mathbf{x}_1 = \left[\begin{array}{ccc} \frac{1}{2} & \frac{1}{3} & \frac{1}{3} \ 0 & \frac{1}{3} & \frac{2}{3} \ \frac{1}{2} & \frac{1}{3} & 0 \end{array}\right] \left[\begin{array}{r} 120 \ 180 \ 90 \end{array}\right]\]Calculate each component:\- \( x_{1,1} = \left(\frac{1}{2} \times 120\right) + \left(\frac{1}{3} \times 180\right) + \left(\frac{1}{3} \times 90\right) = 60 + 60 + 30 = 150 \)- \( x_{1,2} = \left(0 \times 120\right) + \left(\frac{1}{3} \times 180\right) + \left(\frac{2}{3} \times 90\right) = 0 + 60 + 60 = 120 \)- \( x_{1,3} = \left(\frac{1}{2} \times 120\right) + \left(\frac{1}{3} \times 180\right) + \left(0 \times 90\right) = 60 + 60 + 0 = 120 \)The first-step state vector is \( \mathbf{x}_1 = \begin{bmatrix} 150 \ 120 \ 120 \end{bmatrix} \).
03

Calculate Second Step State Vector

Now compute \( \mathbf{x}_2 \), the state vector after two steps, by multiplying \( P \) with \( \mathbf{x}_1 \):\[\mathbf{x}_2 = \left[\begin{array}{ccc} \frac{1}{2} & \frac{1}{3} & \frac{1}{3} \ 0 & \frac{1}{3} & \frac{2}{3} \ \frac{1}{2} & \frac{1}{3} & 0 \end{array}\right] \left[\begin{array}{r} 150 \ 120 \ 120 \end{array}\right]\]Calculate each component:- \( x_{2,1} = \left(\frac{1}{2} \times 150\right) + \left(\frac{1}{3} \times 120\right) + \left(\frac{1}{3} \times 120\right) = 75 + 40 + 40 = 155 \)- \( x_{2,2} = \left(0 \times 150\right) + \left(\frac{1}{3} \times 120\right) + \left(\frac{2}{3} \times 120\right) = 0 + 40 + 80 = 120 \)- \( x_{2,3} = \left(\frac{1}{2} \times 150\right) + \left(\frac{1}{3} \times 120\right) + \left(0 \times 120\right) = 75 + 40 + 0 = 115 \)The second-step state vector is \( \mathbf{x}_2 = \begin{bmatrix} 155 \ 120 \ 115 \end{bmatrix} \).
04

Determine Proportion in State 1

The proportion of the state 1 population still in state 1 after two steps is found by dividing the state 1 value in \( \mathbf{x}_2 \) by the initial number of people in state 1. That is \( \text{Proportion} = \frac{155}{120} \), which simplifies to approximately 1.29 or 129% when expressed as a percentage.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transition Matrix
In the world of Markov chains, the transition matrix is crucial. It represents the probabilities of moving from one state to another in a state space. Each entry in the matrix shows the likelihood of transitioning between states. For our exercise, we look at a matrix called \( P \). This is a 3x3 transition matrix depicting the probabilities across three different states:
\[ P = \begin{bmatrix} \frac{1}{2} & \frac{1}{3} & \frac{1}{3} \ 0 & \frac{1}{3} & \frac{2}{3} \ \frac{1}{2} & \frac{1}{3} & 0 \end{bmatrix} \]
In this matrix:
  • Each row sums to 1. This represents the total probability of being in any state.
  • Values indicate transition likelihoods, such as \( \frac{1}{2} \) meaning a 50% chance.
Understanding the transition matrix is essential for predicting future state distributions.
State Vector
The state vector gives us a snapshot of the current state population distribution. It's not just any vector - it provides quantitative data about where entities are across states at any given time. In our exercise, the initial state vector, \( \mathbf{x}_0 \), is:
\[ \mathbf{x}_0 = \begin{bmatrix} 120 \ 180 \ 90 \end{bmatrix} \]
This represents the number of people in states 1, 2, and 3 initially. Using this vector along with the transition matrix allows us to determine how many people move between states as time progresses. Each component of the vector correlates to a state, creating a bridge between abstract numbers and real-world scenarios.
Matrix Multiplication
Matrix multiplication is at the core of transitioning between states over time in Markov chains. It combines the transition matrix with the state vector to project future states. In our exercise, we need to compute two steps:
  • First step: Multiply \( P \) by \( \mathbf{x}_0 \) to get \( \mathbf{x}_1 \),
    \[ \mathbf{x}_1 = P \cdot \mathbf{x}_0 \]
  • Second step: Multiply \( P \) again by \( \mathbf{x}_1 \) to get \( \mathbf{x}_2 \),
    \[ \mathbf{x}_2 = P \cdot \mathbf{x}_1 \]
The result is a new state vector showing the distribution among states after one or more time periods by progressively applying the multiplication. This mathematical operation is critical in Markov chains for predicting how entities move through the system over time.
Population Distribution
Population distribution is a key outcome we derive from applying Markov chain models, especially after utilizing matrix multiplication. It shows us how populations are likely to shift from one state to another over defined periods. After two transitions in our exercise, the new population distribution among the states is:
\[ \mathbf{x}_2 = \begin{bmatrix} 155 \ 120 \ 115 \end{bmatrix} \]
This vector tells us:
  • 155 people are now in state 1.
  • 120 people remain or move to state 2.
  • 115 people end up in state 3.
The analysis of such distributions helps in planning and adjusting resources based on predicted population movements. Accurate understanding and prediction of such changes can have significant implications in various fields such as economics, social sciences, and more.

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