Problem 2) Consider a Markov chain with states 0, 1, 2 and the following transition probability matrix[1/2 1/3 1/61P =| 0 1/3 2/31/2]li/2If p(Xo = 0) = 0.25 and p(Xo = 1) = 0.25, then find p(X2 = 1).%3D

Answered: Problem 2) Consider a Markov chain with…

A First Course in Probability (10th Edition)

10th Edition

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Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Answer all the following problems 1. A Markov chain Xo, X₁, X₂,... has the transition probability matrix 0 1 2 0.3 0.1 0 0.6 P-1 0.3 0.3 0.4 2 0.4 0.1 0.5 If it si known that the process starts in state Xo = 1, determine the probability Pr{Xo = 1, X₁0, X₂ = 2} if Pr{Xo = 1) = 0.5.
3. A rat is put into the following maze: 3 moves. 1 2 The rat has a probability of 1/4 of starting in any compartment and suppose that the rat chooses a passageway at random when it makes a move from one compartment to another at each time. Let Xn be the compartment occupied by the rat after n (a) Explain why {X₂} is a MC and find the transition matrix P. (b) Explain why the chain is irreducible, aperiodic and positive recurrent. (c) What is the limit of Pn? Explain. (d) Find the probability that the rat is in compartment 3 after two moves. (e) In the long run, how many times that the rat enters in compartment 4 in 100 movements?
1. Consider a Markov chain X matrix = (Xn)n≥0 with state space S [0.2 0.4 0 0.4 0.3 0 0.7 0 0.5 0 0.5 0 0 0.1 0.9 0 (a) Which states are transient and which are recurrent? (b) Is this markov chain irreducible or reducible? = {1, 2, 3, 4} and transition
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] ?
Consider the migration (Markov) matrix A = 0.7 0 0.2 0.1 0.4 0.2 0.2 0.6 0.6 Suppose that, initially, there are 96 residents in location 1, 82 residents in location 2, and 151 residents in location 3. Assume time is measured in years. Find the population in each location after 1 year. Find the population in each location after 2 years.
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 =
Question 3 Consider a discrete time Markov Chain (Xn} on the state space (1, 2, 3) with the transition matrix given as follows: P= 1/2 1/4 1/4 1/3 2/3 1/2 1/2 Then P( X1 = 3, X2 = 2,X3 = 1) is given by 3/16 O 1/12 4/18 14/36 OO

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Problem 2) Consider a Markov chain with states 0, 1, 2 and the following transition probability matrix 1/2 1/3 1/61 P =|0 1/3 2/3 l1/2 1/2] If p(Xo = 0) = 0.25 and p(Xo = 1) = 0.25, then find p(X2 = 1).