1-α BShow that every 2 x 2 stochastic matrix has at least one steady-state vector. Any such matrix can be written in the form P =α 1-Bwhere x and ẞ are constants between 0 and 1. (There are two linearly independent steady-state vectors if a = ß= 0. Otherwise, there isonly one.)If P is a stochastic matrix, then a steady-state vector, or equilibrium vector, for P is a probability vector q such that Pq=q.If a =B=0, then P=1001Notice that Px = x for any vector x in R². Determine two linearly independent steady-state vectors in this case.0(Use a comma to separate answers as needed.)

To Show: Every 2×2 stochastic matrix has atleast one-stady vector .Any such matrix can be written in…

Answered: Show that every 2 x 2 stochastic matrix…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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What is the second row of the matrix A, if A is stochastic? Second row: 0.8 The steady state vector for A is If v= 0.5 [15] 26 then Av approaches as n gets large. A = [0.2 ? 0.5] ?
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if matrix A= [[a,1-b],[1-a,b]] Assume A does not equal the idenituty . what is the steady state vector for matrix A
3. suppose that the manufacturer keeps a spare machine that only is used when the primary machine is being repaired. During a repair day, the spare machine has a probability of 0.1 of breaking down, in which case it is repaired the next day. Denote the state of the system by (x, y), where x and y, respectively, take on the values 1 or 0 depending upon whether the primary machine (x) and the spare machine (y) are operational (value of 1) or not operational (value of 0) at the end of the day. [Hint: Note that (0, 0) is not a possible state.] (a) Construct the (one-step) transition matrix for this Markov chain. (b) Find the expected recurrence time for the state (1, 0).
(4) Consider a mini-web as a dynamical system. Let A be the coefficient matrix of this system and i be a distribution vector. Prove that if Ax = Xã, then X = 1.
If A is a constant matrix, the fundamental matrix of the system x'(t) = Ax(t) is non-constant. O True False
1. Indicate whether each statement is true or false. No justification is necessary.
In the B&K model of Example 18.5-1, suppose that the interarrival time at the checkout area is exponential with mean 5 minutes and that the checkout time per customer is also exponential with mean 10 minutes. Suppose further that will add a fourth counter. Counters 1,2, and 3 will open based on increments of two customers and counter 4 will open when there are 7 or more in the store. (a) The steady-state probabilities, for all . (b) The probability that a fourth counter will be needed. (c) The average number of idle counters.
Complete each of the following sentences by matching each of the letter options (e.g. (A), (B), ...) to a correct corresponding number option (e.g. (1), (2), ...). A) For the matrix LaTeX: M:=\begin{pmatrix}0& -1\\ 1& 0 \end{pmatrix} we have LaTeX: \text{span}(\{I, M, M^{2}, M^{3},....,M^{n},...\}) is equal to (B) Let LaTeX: x,\:x^2 be polynomial functions in LaTeX: \mathcal P_{2}(\mathbb F_2), the vector space of all polynomial functions of degree two or less with coefficients in LaTeX: \mathbb F_2. Then LaTeX: \mathrm{span}(x) is equal to (C) The vector space of all functions over real numbers is a subspace of (D) Let LaTeX: T:V\to W be a linear transformation and LaTeX: S be a subset of LaTeX: V. Then LaTeX: T(\text{span}(S)) is equal to (1) A strict subspace of LaTeX: \mathrm{span}(x,x^{2}) (that is a subspace of LaTeX: \mathrm{span}(x,x^{2}) which is not equal to LaTeX: \mathrm{span}(x,x^{2})). (2) The space of LaTeX: 2\times 2 matrices with real entries and zero…
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Show that every 2 x 2 stochastic matrix has at least one steady-state vector. Any such matrix can be written in the form P = 1-α B a 1-B where & and B are constants between 0 and 1. (There are two linearly independent steady-state vectors if a=B=0. Otherwise, there is only one.)