Answer all the following problems1. A Markov chain Xo, X₁, X₂,... has the transition probability matrix01 20.3 0.10 0.6P-1 0.3 0.3 0.42 0.4 0.1 0.5If it si known that the process starts in state Xo = 1, determine the probabilityPr{Xo = 1, X₁0, X₂ = 2} if Pr{Xo = 1) = 0.5.

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Answered: Answer all the following problems 1. A…

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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(14) The transition matrix for a Markov chain with three states is 0.3 0.5 0.21) 0.6 0.3 0.1 001. For a group of 100 individuals in state I at time 0, the variance of the number of individuals in state 1 at time 3 is:
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2.Initially we have 2 red marbles in Box 1 and 2 white marble in Box 2. At every time step, we take a marble from each box and swap. Let Xt be the number of red marbles in Box 1 at time t. There are 3 states: 0, 1, 2. Let P be the transition matrix for {X;}, what is (P)oo (probability from state 0 to state 0)?
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Q3: The probability parameters of a homogeneous Markov chain are as follows: 0.8 0.8 0.8 C = 0.2 0.1 0.1 0.1 1. 0.1 If the chain is in the third state after 10 steps of the chain, evaluate the probability that it was also in the third state after 5 steps.
4. A Markov chain has transition matrix 6 1 3 4 Given the initial probabilities ø1 = 02 = ¢3 = , find Pr (X1 X2). %3D %3D 1/4
4. Find the stationary distribution of the Markov chain shown below using the reversible property. OFE 1/2 I 2 1/2 1/4 5/12 3 1/3 1/6 7/12 4 1/4 1/8 7/8 5
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3. A discrete Markov model has state space equal to E {0, 1, 2}. The transition probabilities p are independent of r and recorded in the matrix P defined below. .7 .2 .1 P = .1 .6 .3 1 Assume that Lynn is in state 0 at time 0, and that i = .04. (a.) Determine the probability that Lynn is in state 0 at time 3. (b.) Determine the probability that Lynn transitions from state 1 at time 3 to state 2 at time 4.
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] ?
5. consider the example below, where the states are Condition State and the transition matrix is 0 1 23 2 Πο Good as new Operable-minimum deterioration Operable-major deterioration Inoperable and replaced by a good-as-new machine we found that the steady-state probabilities are 2 13' T1= = P = 78314 00 1 0 7 13' HARTNO LELBLINO 1 16 1 1 1 1 16 8 8 0 2 2 = π2 2 13' I3 = 2 13 (a) Find the expected recurrence time for state 0 (i.e., the expected length of time a machine can be used before it must be replaced) by solving a linear system for Moo, M10, 20, and μ30. (b) Find the expected recurrence time for state O directly by the formula Moo 1 πο =
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.

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Answer all the following problems 1. A Markov chain Xo, X₁, X₂,... has the transition probability matrix 12 0.3 0.1 0.3 0.4 2 0.4 0.1 0.5 0 0 0.6 P=10.3 If it si known that the process starts in state Xo = 1, determine the probability Pr{Xo = 1,X₁ = 0, X₂ = 2} if Pr{Xo = 1} = 0.5.