?Determine whether \(P\) is a regular stochastic matrix. a

Determine whether \(A\) is a stochastic matrix. If \(A\) is not stochastic, then explain why not. a. \(A=\left[\begin{array}{ll}0.4 & 0.3 \\ 0.6 & 0.7\end{array}\right]\) b. \(A=\left[\begin{array}{ll}0.4 & 0.6 \\ 0.3 & 0.7\end{array}\right]\) c. \|(A=\left[\begin{array}{lll}1 & \frac{1}{2} & \frac{1}{3} \\ 0 & 0 & \frac{1}{3} \\ 0 & \frac{1}{2} & \frac{1}{3}\end{array}\right]\) d. \(A=\left[\begin{array}{rrr}\frac{1}{3} & \frac{1}{3} & \frac{1}{2} \\ \frac{1}{6} & \frac{1}{3} & -\frac{1}{2} \\ \frac{1}{2} & \frac{1}{3} & 1\end{array}\right]\0 Equation Transcription: [] [] [] [] Text Transcription: A= [0.6 0.7 0.4 0.3] A= [0.6 0.7 0.4 0.6] A= [ 0 1/2 ? 0 0 ? 1 1/2 1/3] A= [ 1/2 1/3 1 1/6 1/3 - ½ 1/3 1/2 1/3]

Determine whether \(A\) is a stochastic matrix. If \(A\) is not stochastic, then explain why not. a. \(A=\left[\begin{array}{ll}0.2 & 0.9 \\ 0.8 & 0.1\end{array}\right]\) b. \(A=\left[\begin{array}{ll}0.2 & 0.8 \\ 0.9 & 0.1\end{array}\right]\) c. \(A=\left[\begin{array}{ccc}\frac{1}{12} & \frac{1}{9} & \frac{1}{6} \\ \frac{1}{2} & 0 & \frac{5}{6} \\ \frac{5}{12} & \frac{8}{9} & 0\end{array}\right]\) d. \(A=\left[\begin{array}{rrr}-1 & \frac{1}{3} & \frac{1}{2} \\ 0 & \frac{1}{3} & \frac{1}{2} \\ 2 & \frac{1}{3} & 0\end{array}\right]\) Equation Transcription: [] [] [] [] Text Transcription: A= [0.8 0.1 0.2 0.9] A= [0.9 0.1 0.2 0.8] A= [ 5/12 8/9 0 1/2 0 ? 1/12 1/9 1/6] A= [ 2 1/3 0 0 1/3 ½ -1 1/3 1/2]

Use Formulas (11) and (12) to compute the state vector in two different ways. Equation Transcription: [] Text Transcription:

Use Formulas (11) and (12) to compute the state vector \(x_{4} \\\) in two different ways. \(P=\left[\begin{array}{ll}0.8 & 0.5 \\0.2 & 0.5\end{array}\right]\); \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\0\end{array}\right]\) Equation Transcription: [] [] Text Transcription: x_4 P = [0.2 0.5 0.8 0.5] x_0 = [0 1]

Determine whether \(P\) is a regular stochastic matrix. a. \(P=\left[\begin{array}{ll}\frac{1}{5} & \frac{1}{7} \\\frac{4}{5} & \frac{6}{7}\end{array}\right] b. \(P=\left[\begin{array}{ll}\frac{1}{5} & 0 \\\frac{4}{5} & 1\end{array}\right]\) C. \(P=\left[\begin{array}{cc}\frac{1}{5} & 1 \\\frac{4}{5} & 0\end{array}\right]\) Equation Transcription: [] [] [] Text Transcription: P P = [4/5 6/7 1/5 1/7] P = [4/5 1 1/5 0] P = [4/5 0 1/5 1]

Determine whether \(P\) is a regular stochastic matrix. a. \(P=\left[\begin{array}{ll}\frac{1}{2} & 1 \\\frac{1}{2} & 0\end{array}\right]\) b. \(P=\left[\begin{array}{ll}1 & \frac{2}{3} \\0 & \frac{1}{3}\end{array}\right]\) c. \(P=\left[\begin{array}{ll}\frac{3}{4} & \frac{1}{3} \\\frac{1}{4} & \frac{2}{3}\end{array}\right]\) Equation Transcription: [] [] [] Text Transcription: P P = [2 0 1/2 1] P = [0 ? 1 2 /3] P = [1/4 ? 3/4 1/3]

Verify that \(P\) is a regular stochastic matrix, and find the steady-state vector for the associated Markov chain. \(P=\left[\begin{array}{ll}\frac{1}{4} & \frac{2}{3} \\\frac{3}{4} & \frac{1}{3}\end{array}\right]\) Equation Transcription: [] Text Transcription: P P = [3/4 ? 1/4 2/3]

Verify that \(P\) is a regular stochastic matrix, and find the steady-state vector for the associated Markov chain. \(P=\left[\begin{array}{ll}0.2 & 0.6 \\0.8 & 0.4\end{array}\right]\) Equation Transcription: [] Text Transcription: P P = [0.8 0.4 0.2 0.6]

Verify that \(P\) is a regular stochastic matrix, and find the steady-state vector for the associated Markov chain. \(P=\left[\begin{array}{lll}\frac{1}{2} & \frac{1}{2} & 0 \\\frac{1}{4} & \frac{1}{2} & \frac{1}{3} \\\frac{1}{4} & 0 & \frac{2}{3}\end{array}\right]\) Equation Transcription: [] Text Transcription: P P P = [1/4 0 ? 1/4 1/2 1/3 1/2 1/2 0 ]

Verify that \(P\) is a regular stochastic matrix, and find the steady-state vector for the associated Markov chain. \(P=\left[\begin{array}{ccc}\frac{1}{3} & \frac{1}{4} & \frac{2}{5} \\0 & \frac{3}{4} & \frac{2}{5} \\\frac{2}{3} & 0 & \frac{1}{5}\end{array}\right] Equation Transcription: [] Text Transcription: P P = [2/3 0 ? 0 3/4 2/5 13/ 1/4 2/5 ]

Consider a Markov process with transition matrix a. What does the entry 0.2 represent? b. What does the entry 0.1 represent? c. If the system is in state 1 initially, what is the probability that it will be in state 2 at the next observation? d. If the system has a 50% chance of being in state 1 initially, what is the probability that it will be in state 2 at the next observation?

Consider a Markov process with transition matrix a. What does the entry represent? b. What does the entry 0 represent? c. If the system is in state 1 initially, what is the probability that it will be in state 1 at the next observation? d. If the system has a 50% chance of being in state 1 initially, what is the probability that it will be in state 2 at the next observation? Equation Transcription: Text Transcription: 6/7

On a given day the air quality in a certain city is either good or bad. Records show that when the air quality is good on one day, then there is a 95% chance that it will be good the next day, and when the air quality is bad on one day, then there is a 45% chance that it will be bad the next day. a. Find a transition matrix for this phenomenon. b. If the air quality is good today, what is the probability that it will be good two days from now? c. If the air quality is bad today, what is the probability that it will be bad three days from now? d. If there is a 20% chance that the air quality will be good today, what is the probability that it will be good tomorrow?

In a laboratory experiment, a mouse can choose one of two food types each day, type I or type II. Records show that if the mouse chooses type I on a given day, then there is a 75% chance that it will choose type I the next day, and if it chooses type II on one day, then there is a 50% chance that it will choose type II the next day. a. Find a transition matrix for this phenomenon. b. If the mouse chooses type I today, what is the probability that it will choose type I two days from now? c. If the mouse chooses type II today, what is the probability that it will choose type II three days from now? d. If there is a 10% chance that the mouse will choose type I today, what is the probability that it will choose type I tomorrow?

Suppose that at some initial point in time 100,000 people live in a certain city and 25,000 people live in its suburbs. The Regional Planning Commission determines that each year 5% of the city population moves to the suburbs and 3% of the suburban population moves to the city. a. Assuming that the total population remains constant, make a table that shows the populations of the city and its suburbs over a five-year period (round to the nearest integer). b. Over the long term, how will the population be distributed between the city and its suburbs?

Suppose that two competing television stations, station 1 and station 2, each have 50% of the viewer market at some initial point in time. Assume that over each one-year period station 1 captures 5% of station 2’s market share and station 2 captures 10% of station 1’s market share. a. Make a table that shows the market share of each station over a five-year period. b. Over the long term, how will the market share be distributed between the two stations?

Fill in the missing entries of the stochastic matrix \(P=\left[\begin{array}{ccc}\frac{7}{10} & * & \frac{1}{5} \\* & \frac{3}{10} & * \\\frac{1}{10} &\frac{3}{5} & \frac{3}{10}\end{array}\right]\) and find its steady-state vector. Equation Transcription: [] Text Transcription: P = [ 1/10 3 5 3 10 * 3/10 * 7/10 * 1/5]

If \(P\) is an \(nxn\)stochastic matrix, and if \(M\) is a \(1xn\) matrix whose entries are all 1 's, then \(MP=\)_______________ Equation Transcription: Text Transcription: P N xn M 1xn MP=

If \(P\) is a regular stochastic matrix with steady-state vector \(q\), what can you say about the sequence of products \(P q, P^{2} q, P^{3} q, \ldots, P^{k} q, \ldots\) as \(k \rightarrow \infty \\\)? Equation Transcription: Text Transcription: P q Pq, P^2q, P^3q,…, P^kq,… k rightarrow infty

a. If \(P\) is a regular \(n \times n \\\) stochastic matrix with steady-state vector \(q\), and if \(e_{1}, e_{2}, \ldots, e_{n} \\\) are the standard unit vectors in column form, what can you say about the behavior of the sequence \(P e_{i}, P^{2} e_{i}, P^{3} e_{i}, \ldots, P^{k} e_{i}, \ldots \\\) As \(k \rightarrow \infty \\\) for each \(i=1,2, \ldots, n \\\)? b. What does this tell you about the behavior of the column vectors of \(P^{k}\) as \(k \rightarrow \infty \\\)? Equation Transcription: Text Transcription: P n x n q e_1,e_2,…,e_n Pe_i, P^2e_i, P^3e_i,…, P^ke_i,… K?? i=1,2,…,n P^k

Prove that the product of two stochastic matrices with the same size is a stochastic matrix. [Hint: Write each column of the product as a linear combination of the columns of the first factor.]

Prove that if \(P\) is a stochastic matrix whose entries are all greater than or equal to \(\rho\), then the entries of \(P^{2}\) are greater than or equal to \(\rho\). Equation Transcription: Text Transcription: P P^2 rho

TF. In parts (a)–(g) determine whether the statement is true or false, and justify your answer. The vector \(\left[\begin{array}{l}\frac{1}{3} \\0 \\\frac{2}{3}\end{array}\right]\) is a probability vector. Equation Transcription: [] Text Transcription: [2/3 0 1/3]

TF. In parts (a)–(g) determine whether the statement is true or false, and justify your answer. The matrix \(\left[\begin{array}{ll}0.2 & 1 \\0.8 & 0\end{array}\right]\) is a regular stochastic matrix. Equation Transcription: [] Text Transcription: [ 0.8 0 0.2 1]

TF. In parts (a)–(g) determine whether the statement is true or false, and justify your answer. The column vectors of a transition matrix are probability vectors.

TF. In parts (a)–(g) determine whether the statement is true or false, and justify your answer. A steady-state vector for a Markov chain with transition matrix \(P\) is any solution of the linear system \((I-P) q=0\). Equation Transcription: Text Transcription: P (I-P) q=0

TF. In parts (a)–(g) determine whether the statement is true or false, and justify your answer. The square of every regular stochastic matrix is stochastic.

TF. In parts (a)–(g) determine whether the statement is true or false, and justify your answer. A vector with real entries that sum to 1 is a probability vector.

TF. In parts (a)–(g) determine whether the statement is true or false, and justify your answer. Every regular stochastic matrix has \(\lambda=1\) as an eigenvalue. Equation Transcription: Text Transcription: lambda=1

In Examples 4 and 9 we considered the Markov chain with transition matrix ???? and initial state vector x(0) where \(P=\left[\begin{array}{lll}0.5 & 0.4 & 0.6 \\0.2 & 0.2 & 0.3 \\0.3 & 0.4 & 0.1\end{array}\right]\) and \(\mathbf{x}(0)=\left[\begin{array}{l}0 \\1 \\0\end{array}\right]\) a. Confirm the numerical values of \(x(1), x(2)\), . . . , \(x(6)\) obtained in Example 4 using the method given in that example. b. As guaranteed by part (c) of Theorem 5.5.1, confirm that the sequence \(P\), \(P^{2}\), ….\(P^{k}\),.....converges to the matrix \(Q\) each of whose column vectors is the steady-state vector \(q\) obtained in Example Equation Transcription: [] [] Text Transcription: P x (0) P = [0.3 0.4 0.1 0.2 0.2 0.3 0.5 0.4 0.6 ] x (0) = [0 1 0] x (1) x (2) x (6) P P^2 P^k Q q

Suppose that a car rental agency has three locations, numbered 1, 2, and 3. A customer may rent a car from any of the three locations and return it to any of the three locations. Records show that cars are rented and returned in accordance with the following probabilities: a. Assuming that a car is rented from location 1, what is the probability that it will be at location 1 after two rentals? b. Assuming that this dynamical system can be modeled as a Markov chain, find the steady-state vector. c. If the rental agency owns 120 cars, how many parking spaces should it allocate at each location to be reasonably certain that it will have enough spaces for the cars over the long term? Explain your reasoning.

Physical traits are determined by the genes that an offspring receives from its parents. In the simplest case a trait in the offspring is determined by one pair of genes, one member of the pair inherited from the male parent and the other from the female parent. Typically, each gene in a pair can assume one of two forms, called alleles, denoted by ???? and a. This leads to three possible pairings: ????????, ????a, aa called genotypes (the pairs ????a and a???? determine the same trait and hence are not distinguished from one another). It is shown in the study of heredity that if a parent of known genotype is crossed with a random parent of unknown genotype, then the offspring will have the genotype probabilities given in the following table, which can be viewed as a transition matrix for a Markov process: Thus, for example, the offspring of a parent of genotype ???????? that is crossed at random with a parent of unknown genotype will have a 50% chance of being ????????, a 50% chance of being ????a, and no chance of being aa. a. Show that the transition matrix is regular. b. Find the steady-state vector and discuss its physical interpretation.