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Chapter 7: Problem 22

You are given a transition matrix \(P\) and initial distribution vector \(v\). Find (a) the two-step transition matrix and (b) the distribution vectors after one, two, and three steps. $$ P=\left[\begin{array}{ccc} 1 / 2 & 1 / 2 & 0 \\ 1 / 2 & 1 / 2 & 0 \\ 1 / 2 & 0 & 1 / 2 \end{array}\right], v=\left[\begin{array}{lll} 0 & 0 & 1 \end{array}\right] $$

Short Answer

Expert verified
(a) The two-step transition matrix is \[ P^2 = \left[\begin{array}{ccc} 1/2 & 1/2 & 0\\ 1/2 & 1/2 & 0\\ 1/2 & 1/4 & 1/4 \end{array}\right] \] (b) The distribution vectors after one, two, and three steps are: - After one step: [0, 0, 1/2] - After two steps: [0, 0, 1/4] - After three steps: [0, 0, 1/8]

Step by step solution

01

Calculate the one-step transition matrix and the distribution vector after one step

Multiply the transition matrix P by the initial distribution vector v as follows: \(Pv = Av\) \[ \left[\begin{array}{ccc} 1 / 2 & 1 / 2 & 0 \\ 1 / 2 & 1 / 2 & 0 \\ 1 / 2 & 0 & 1 / 2 \end{array}\right] \left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \] Now, let's perform the matrix multiplication to obtain the distribution vector after one step: \[ Av = \left[\begin{array}{c} (1/2 \cdot 0) + (1/2 \cdot 0) + (0 \cdot 1) \\ (1/2 \cdot 0) + (1/2 \cdot 0) + (0 \cdot 1) \\ (1/2 \cdot 0) + (0 \cdot 0) + (1/2 \cdot 1) \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 1/2 \end{array}\right] \] So, after one step, the distribution vector is [0, 0, 1/2].
02

Calculate the two-step transition matrix

Compute the product of the transition matrix P with itself to find the two-step transition matrix: \[P^2 = PP\] \[ \left[\begin{array}{ccc} 1 / 2 & 1 / 2 & 0 \\ 1 / 2 & 1 / 2 & 0 \\ 1 / 2 & 0 & 1 / 2 \end{array}\right] \left[\begin{array}{ccc} 1 / 2 & 1 / 2 & 0 \\ 1 / 2 & 1 / 2 & 0 \\ 1 / 2 & 0 & 1 / 2 \end{array}\right] \] After performing the matrix multiplication, we get: \[ P^2 = \left[\begin{array}{ccc} 1/2 & 1/2 & 0\\ 1/2 & 1/2 & 0\\ 1/2 & 1/4 & 1/4 \end{array}\right] \]
03

Calculate the distribution vectors after two and three steps

Now that we have obtained the two-step transition matrix, we can calculate the distribution vector after two steps by multiplying P^2 by the initial distribution vector v: \[P^2v = Bv\] \[ \left[\begin{array}{ccc} 1/2 & 1/2 & 0\\ 1/2 & 1/2 & 0\\ 1/2 & 1/4 & 1/4 \end{array}\right] \left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \] Perform the matrix multiplication: \[ Bv = \left[\begin{array}{c} (1/2 \cdot 0) + (1/2 \cdot 0) + (0 \cdot 1) \\ (1/2 \cdot 0) + (1/2 \cdot 0) + (0 \cdot 1) \\ (1/2 \cdot 0) + (1/4 \cdot 0) + (1/4 \cdot 1) \end{array}\right] =\left[\begin{array}{c} 0 \\ 0 \\ 1/4 \end{array}\right] \] So, after two steps, the distribution vector is [0, 0, 1/4]. For the distribution vector after three steps, we need to multiply the two-step transition matrix P^2 by the distribution vector obtained after one step (Av): \[P^2(Av) = Cv\] \[ \left[\begin{array}{ccc} 1/2 & 1/2 & 0\\ 1/2 & 1/2 & 0\\ 1/2 & 1/4 & 1/4 \end{array}\right] \left[\begin{array}{c} 0 \\ 0 \\ 1/2 \end{array}\right] \] Perform the matrix multiplication: \[ Cv = \left[\begin{array}{c} (1/2 \cdot 0) + (1/2 \cdot 0) + (0 \cdot 1/2) \\ (1/2 \cdot 0) + (1/2 \cdot 0) + (0 \cdot 1/2) \\ (1/2 \cdot 0) + (1/4 \cdot 0) + (1/4 \cdot 1/2) \end{array}\right] =\left[\begin{array}{c} 0 \\ 0 \\ 1/8 \end{array}\right] \] So, after three steps, the distribution vector is [0, 0, 1/8]. To summarize: (a) The two-step transition matrix is \[ \left[\begin{array}{ccc} 1/2 & 1/2 & 0\\ 1/2 & 1/2 & 0\\ 1/2 & 1/4 & 1/4 \end{array}\right] \] (b) The distribution vectors are as follows: - After one step: [0, 0, 1/2] - After two steps: [0, 0, 1/4] - After three steps: [0, 0, 1/8]

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Key Concepts

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

Distribution Vector
When learning about stochastic processes, understanding the distribution vector is crucial. Essentially, a distribution vector represents the probabilities that a system will be in a particular state at a given time. In the context of our exercise, the initial distribution vector, denoted here as v, shows the starting condition of the system. Each element of this vector corresponds to a state in the system, and the probabilities in it must sum up to 1.

In our exercise, the initial distribution vector is v = [0, 0, 1], indicating that the system is initially in the third state with certainty. As we use matrix multiplication to evolve the distribution vector through time under the transition matrix P, we keep track of how the probability of being in each state changes. This evolution is what gives us new distribution vectors after each step, providing insight into the system's dynamics at future steps.
Matrix Multiplication
Matrix multiplication may seem daunting at first, but it is a fundamental operation in linear algebra used to solve a wide range of problems, including those in Markov processes. In our exercise, matrix multiplication is the method through which we update the distribution vector in each step. Here's a simplified breakdown:

When you multiply a matrix, such as the transition matrix P, by a vector, such as v, you calculate the resulting vector by taking the dot product of the rows of the matrix with the column vector. For instance, the transition matrix P has three rows and three columns, and when you multiply it by the initial distribution vector v with three rows, you end up with a new vector which is a linear combination of the columns of P, weighted by the elements of v.

It is essential to remember that the order of multiplication is important and not all matrices can be multiplied together; they must be compatible in size, specifically the number of columns in the first matrix must match the number of rows in the second matrix.
Markov Processes
The concept of Markov processes is pivotal in understanding systems that go through transitions with probabilities that only depend on the current state, not the history of past states. This idea is known by the term 'memorylessness' and is core to Markov processes.

The transition matrix P, used in our exercise, is a cornerstone of a Markov process. It describes the likelihood of moving from one state to another in one step. In each row of matrix P, the sum of the probabilities is 1, reflecting that from any given state, the system must transition to some state in the next step.

In applying the transition matrix successively through matrix multiplication, we forecast the system's behavior over time. With each step, we gain insights into the probabilities of being in each state, thus understanding the evolution of the Markov process. This stepwise evaluation illustrates the iterative nature of Markov processes and how the state's probability distribution evolves over time.

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Most popular questions from this chapter

Auto Sales In April 2008, the probability that a randomly chosen new automobile was manufactured by Ford was \(.15\), while the probability that it was manufactured by Chrysler was .12. \({ }^{35}\) What is the probability that a randomly chosen new automobile was manufactured by neither company?

Explain: If \(Q\) is a matrix whose rows are steady-state distribution vectors, then \(Q P=Q\).

Based on the following table, which shows the performance of a selection of 100 stocks after one year. (Take S to be the set of all stocks represented in the table.) $$ \begin{array}{|r|c|c|c|c|} \hline & \multicolumn{3}{|c|} {\text { Companies }} & \\ \cline { 2 - 4 } & \begin{array}{c} \text { Pharmaceutical } \\ \boldsymbol{P} \end{array} & \begin{array}{c} \text { Electronic } \\ \boldsymbol{E} \end{array} & \begin{array}{c} \text { Internet } \\ \boldsymbol{I} \end{array} & \text { Total } \\ \hline \begin{array}{r} \text { Increased } \\ \boldsymbol{V} \end{array} & 10 & 5 & 15 & 30 \\ \hline \begin{array}{r} \text { Unchanged }^{*} \\ \boldsymbol{N} \end{array} & 30 & 0 & 10 & 40 \\ \hline \begin{array}{r} \text { Decreased } \\ \boldsymbol{D} \end{array} & 10 & 5 & 15 & 30 \\ \hline \text { Total } & 50 & 10 & 40 & 100 \\ \hline \end{array} $$ If a stock stayed within \(20 \%\) of its original value, it is classified as "unchanged." Use symbols to describe the event that an Internet stock did not increase. How many elements are in this event?

Employment You have worked for the Department of Administrative Affairs (DAA) for 27 years, and you still have little or no idea exactly what your job entails. To make your life a little more interesting, you have decided on the following course of action. Every Friday afternoon, you will use your desktop computer to generate a random digit from 0 to 9 (inclusive). If the digit is a zero, you will immediately quit your job, never to return. Otherwise, you will return to work the following Monday. a. Use the states (1) employed by the DAA and (2) not employed by the DAA to set up a transition probability matrix \(P\) with decimal entries, and calculate \(P^{2}\) and \(P^{3}\). b. What is the probability that you will still be employed by the DAA after each of the next three weeks? c. What are your long-term prospects for employment at the DAA? HIIIT [See Example 5.]

Social Security According to the New York Times/CBS poll of March, 2005, referred to in Exercise \(65,49 \%\) agreed that Social Security taxes should be raised if necessary to keep the system afloat, and \(43 \%\) agreed that it would be a good idea to invest part of their Social Security taxes on their own. \({ }^{38} \mathrm{What}\) is the largest percentage of people who could have agreed with at least one of these statements? What is the smallest percentage of people who could have agreed with at least one of these statements?
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