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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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
why is the covariance of a deterministic and a stochastic process 0? This relats to Arithmetic Bronian Motion
Answer all three subparts
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 =
The figure above illustrates a continuous-time Markov process. Suppose the system is currently instate 2. After a small amount of time Δ, what is the probability that the system is in each state?
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.
Given the transition matrix of a Markov process, what is the probability of going to state 3 from state 1 after 3 steps?
A Markov chain has the transition matrix shown below: 0.2 0.1 0.7] P = 0.6 0.4 1 (Note: For questions 1, 3, and 5, express your answers as decimal fractions rounded to 4 decimal places (if they have more than 4 decimal places).) (1) If, on the first observation the system is in state 1, what is the probability that it is in state 1 on the next observation? (2) If, on the first observation the system is in state 1, what state is the system most likely to occupy on the next observation? (If there is more than one such state, which is the first one.) (3) If, on the first observation, the system is in state 1, what is the probability that the system is in state 1 on the third observation? ...

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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.