16. GivenA =1 0 00 0.7 0.20 0.3 0.83(a) Is the matrix A regular?(b) Does the limit limk→ Ak exist? If so, find the limit.(c) Find the limit for a Markov chain with initial probability vector P = (0.500.5)and the matrix A as the transition matrix.

The given matrix is A=10000.70.200.30.8 (a) To check if the matrix is regular. (b) to find the limit…

Answered: 16. Given A = 0 0 0 0.7 0.2 0 0.3 0.8 3…

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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A Markov chain has the transition matrix shown below: [0.4 0.3 0.3] P=0.8 0.2 0 00 1 (Note: Express your answers as decimal fractions rounded to 4 decimal places (if they have more than 4 decimal places).) (1) Find the two-step transition matrix P(2) = [ (2) Find the three-step transition matrix P(3) = (3) Find the three-step transition probability p32(3) =
• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain starts in state x, Pμ(A) the probability when the initial state is random with distribution μ. Ty = min{n ≥ 1: X₂ = y} is the first time after 0 that the chain visits state y. Px,y = Px(Ty < ∞) . Ny is the number of visits to state y after time 0. 4. Let Sn, n20 be a random walk on Z with step distribution 1-p 2 P 1) = 1/², P(X₁ = 1) = 2/₁ 2' 2' P(X₂ = 0) P(X₂ = − 1) = for some 0 < p < 1, p ‡ / . We may denote q=1 - p. That is, the increments (X₂)₁ are i.i.d. and S₂ = X₁ + … + Xn for n ≥ 1 and S = 0. Compute E[S2+1 Sn] for n ≥ 1. (b) Show that Mn = (q/p) Sn defines a martingale (with respect to (Xk)k-1). (c) Does the limit limn→∞ Mn exist almost surely? If yes, give a justification. If your answer is no, explain why. (d) Let T be the first time that S is equal to either −3 or 3. Compute P(ST = 3). Hint: You may use, without proof, the fact P(T<∞) = 1 and that the Optional Stopping Theorem…
Suppose qo = [1/4, 3/4] is the initial state distribution for a Markov process with the following transition matrix: %3D [1/2 1/2] M = [3/4 1/4] (a) Find q1, 92, and q3. (b) Find the vector v that qo M" approaches. (c) Find the matrix that M" approaches.
If she made the last free throw, then her probability of making the next one is 0.7. On the other hand, If she missed the last free throw, then her probability of making the next one is 0.2. Assume that state 1 is Makes the Free Throw and that state 2 is Misses the Free Throw. (1) Find the transition matrix for this Markov process. %3D
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Recall that the graph of a discrete-time, finite-state, homogeneous Markov chain on two states 1, 2 with transition matrix is CO (: :) 1 T = If instead a new Markov chain on three states {1, 2, 3} has transition matrix 10 1 0 (a) Draw the corresponding graph for this new Markov chain. (b) Is the Markov chain with transition matrix T irreducible?
Consider the following. B = {(4,-1,-4), (−3, 1, 4), (-6, 2, 10)}, B' = {(-3, -2, -24), (2, 1, 12), (4, 2, 28)}, 2 1 3 1 (a) Find the transition matrix from B to B'. [X]B' = p-1 = (b) Find the transition matrix from B' to B. P = (c) Verify that the two transition matrices are inverses of each other. E PP-1 =
Suppose that a Markov chain with 3 states and with transition matrix P is in state 3 on the first observation. Which of the following expressions represents the probability that it will be in state 1 on the third observation? (A) the (3, 1) entry of P3 (B) the (1,3) entry of P3 (C) the (3, 1) entry of Pª (D) the (1,3) entry of P2 (E) the (3, 1) entry of P (F) the (1,3) entry of P4 (G) the (3, 1) entry of P2 (H) the (1,3) entry of P
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Consider the following. B = {(-2, -12, -6), (1, 4, 2), (3, 12, 7)}, B' = {(−1, 1, 3), (-2, 1, 3), (-2, 1, 4)}, [x] B¹ (a) Find the transition matrix from B to B'. P-1 = P = 2 (b) Find the transition matrix from B' to B. PP-1 = -26, 9, 27 -16, 5, 14 -10, -29 (c) Verify that the two transition matrices are inverses of each other. [x] B = ↓ 1 (d) Find the coordinate matrix [x]B, given the coordinate matrix [×]B'.
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Consider the following. B = {(-2, 12, -9), (-1, 4, -3), (-3, 12, -8)), B' = {(3, 2, 4), (1, 1, 2), (2, 2, 5)}, [x]g' = 2 (a) Find the transition matrix from B to B'. p-1 = (b) Find the transition matrix from B' to B. P = (c) Verify that the two transition matrices are inverses of each other. pp-1 = 000 (d) Find the coordinate matrix [x], given the coordinate matrix [x]g'. [x] B = ↓↑

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16. Given A = 0 0 0 0.7 0.2 0 0.3 0.8 3 (a) Is the matrix A regular? (b) Does the limit limk→ Ak exist? If so, find the limit. (c) Find the limit for a Markov chain with initial probability vector P = (0.500.5) and the matrix A as the transition matrix.