/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Find \(X_{2}\) (the probability ... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 16

Find \(X_{2}\) (the probability distribution of the system after two observations) for the distribution vector \(X_{0}\) and the transition matrix \(T\). \(X_{0}=\left[\begin{array}{l}\frac{1}{2} \\ \frac{1}{2} \\\ 0\end{array}\right], T=\left[\begin{array}{lll}\frac{1}{2} & \frac{1}{3} & \frac{1}{2} \\ 0 & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4}\end{array}\right]\)

Short Answer

Expert verified
After two observations, the probability distribution vector of the system is \(X_2 = \left[\begin{array}{lll}\frac{1}{4} & \frac{5}{36} & \frac{11}{80} \end{array}\right]\).

Step by step solution

01

Calculate the distribution after the first observation (饾憢鈧�)

To compute the distribution vector after the first observation, we need to multiply the initial distribution vector \(X_0\) with the transition matrix \(T\): \(X_1 = X_0 \cdot T\) \[X_1 = \left[\begin{array}{l}\frac{1}{2} \\\ \frac{1}{2} \\\ 0\end{array}\right] \cdot \left[\begin{array}{lll}\frac{1}{2} & \frac{1}{3} & \frac{1}{2} \\\ 0 & \frac{1}{3} & \frac{1}{4} \\\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4}\end{array}\right]\] Perform the matrix multiplication: \[X_1 = \left[\begin{array}{lll}\frac{1}{4}+\frac{0}{1/2}+\frac{0}{1/2} & \frac{1}{6}+\frac{1}{2} & \frac{1}{4}+\frac{0}{1/4} \end{array}\right]\] \[X_1 = \left[\begin{array}{lll}\frac{1}{4} & \frac{1}{5} & \frac{1}{4} \end{array}\right]\]
02

Calculate the distribution after the second observation (饾憢鈧�)

Now, to compute the distribution vector after the second observation, we need to multiply the distribution vector \(X_1\) we just found with the transition matrix \(T\) once more: \(X_2 = X_1 \cdot T\) \[X_2 = \left[\begin{array}{lll}\frac{1}{4} & \frac{1}{5} & \frac{1}{4} \end{array}\right] \cdot \left[\begin{array}{lll}\frac{1}{2} & \frac{1}{3} & \frac{1}{2} \\\ 0 & \frac{1}{3} & \frac{1}{4} \\\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4}\end{array}\right]\] Perform the matrix multiplication: \[X_2 = \left[\begin{array}{lll}\frac{1}{8}+\frac{0}{1/5}+\frac{1}{8} & \frac{1}{12}+\frac{1}{9}+\frac{1}{12} & \frac{1}{8}+\frac{1}{20}+\frac{1}{16} \end{array}\right]\] \[X_2 = \left[\begin{array}{lll}\frac{1}{4} & \frac{5}{36} & \frac{11}{80} \end{array}\right]\] So after two observations, the probability distribution vector of the system is \(X_2 = \left[\begin{array}{lll}\frac{1}{4} & \frac{5}{36} & \frac{11}{80} \end{array}\right]\).

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Key Concepts

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

Finite Mathematics
Finite Mathematics is an area of mathematics that deals with mathematical concepts and techniques that are used in real-world applications and fields such as business, economics, social sciences, and biological sciences. Unlike calculus, which deals with continuous aspects, finite mathematics focuses on discrete elements. It involves topics like probability and statistics, matrices, linear programming, and finance.

In the context of our example, we apply finite mathematics to analyze probability distributions through the use of matrices. Matrices are particularly useful in representing and solving problems related to transitions between states in a system, such as the likelihood of moving from one state to another in a Markov process. Understanding the basic principles of finite mathematics allows us to interpret the results of matrix operations in terms of real-world scenarios, such as predicting weather patterns, analyzing consumer behavior, or modeling stock market trends.
Transition Matrix
A transition matrix, in the study of probability, is a square matrix that describes the transitions from one state to another in a Markov chain. Each element in the matrix represents the probability of moving from one state to a different state in one time step.

To qualify as a transition matrix, it must have certain properties: the elements are non-negative, and the sum of each row must be 1, as they represent probabilities. In our example, the transition matrix is used to calculate the probability distribution of a system after one or more observations. Understanding how to use and interpret a transition matrix is essential when modeling systems that evolve over time with probabilities that depend only on the current state, not on how the system arrived at that state.
Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra where two matrices are combined to produce a third matrix. This is not a simple element-wise operation but involves a series of computations. In the multiplication of two matrices, the element in the ith row and jth column of the resulting matrix is the sum of the products of corresponding elements from the ith row of the first matrix and the jth column of the second matrix.

This form of multiplication is used to combine transformations or to find subsequent probability distributions in a Markov chain, as seen in the provided exercise. The result of the multiplication provides a new matrix which encapsulates compounded effects of the two originals on a system or scenario being modeled. For students, mastering matrix multiplication is crucial for solving an array of problems within finite mathematics and beyond.

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Most popular questions from this chapter

The Maxwells have decided to invest \(\$ 40,000\) in the common stocks of two companies listed on the New York Stock Exchange. One of the companies derives its revenue mainly from its worldwide operation of a chain of hotels, whereas the other company is a domestic major brewery. It is expected that if the economy is in a state of growth, then the hotel stock should outperform the brewery stock; however, the brewery stock is expected to hold its own better than the hotel stock in a recessionary period. Suppose the following payoff matrix gives the expected percentage increase or decrease in the value of each investment for each state of the economy: $$ \begin{array}{l} \text { Investment in hotel stock } \\ \text { Investment in brewery stock } \end{array}\left[\begin{array}{cc} 25 & -5 \\ 10 & 15 \end{array}\right] $$ a. Determine the optimal investment strategy for the Maxwells' investment of \(\$ 40,000\). b. What profit can the Maxwells expect to make on their investments if they use their optimal investment strategy?

Find the expected payoff \(E\) of each game whose payoff matrix and strategies \(P\) and \(Q\) (for the row and column players, respectively) are given. \(\left[\begin{array}{rr}-1 & 4 \\ 3 & -2\end{array}\right], P=\left[\begin{array}{ll}.8 & .2\end{array}\right], Q=\left[\begin{array}{l}.6 \\\ .4\end{array}\right]\)

Rewrite each absorbing stochastic matrix so that the absorbing states appear first, partition the resulting matrix, and identify the submatrices \(R\) and \(S\). \(\left[\begin{array}{rrrr}.4 & .2 & 0 & 0 \\ .2 & .3 & 0 & 0 \\ 0 & .3 & 1 & 0 \\ .4 & .2 & 0 & 1\end{array}\right]\)

The registrar of Computronics Institute has compiled the following statistics on the progress of the school's students in their 2 -yr computer programming course leading to an associate degree: Of beginning students in a particular year, \(75 \%\) successfully complete their first year of study and move on to the second year, whereas \(25 \%\) drop out of the program; of second-year students in a particular year, \(90 \%\) go on to graduate at the end of the year, whereas \(10 \%\) drop out of the program. a. Construct the transition matrix associated with this Markov process. b. Compute the steady-state matrix. c. Determine the probability that a beginning student enrolled in the program will complete the course successfully.

Rewrite each absorbing stochastic matrix so that the absorbing states appear first, partition the resulting matrix, and identify the submatrices \(R\) and \(S\). \(\left[\begin{array}{llll}.1 & 0 & 0 & 0 \\ .2 & 1 & 0 & .2 \\ .3 & 0 & 1 & 0 \\ .4 & 0 & 0 & .8\end{array}\right]\)
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