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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 17EQ

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Let there be r empty urns, where r is a positive integer, and consider a sequence of
independent trials, each consisting of placing a marble in an urn chosen at random. Let
Xn be the number of empty urns after n trials, n = 1; 2; :::.
(a) Briey argue that (Xn}n>1 is a Markov chain, and write down its state space S.
(b) Find the transition probability P(Xn+1 = j]Xn = i) for all i; jES and n = 1; 2; 3.
Is the Markov chain time homogeneous?
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Consider a Markov chain {Xn}n≥0 having the following transition diagram: For this chain, there are two recurrent classes R1 = {6, 7} and R2 = {1, 2, 5}, and one transient class R3 = {3, 4}. Find the period of state Find f33 and f22. Starting at state 3, find the probability that the chain is absorbed into R1. Starting at state 3, find the mean absorbation time, i.e., the expected number of steps that the chain is absorbed into R1 or R2. Note: there are missing transition probabilities for this chain, but no impact for your solution.
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Consider the problem of sending a binary message, 0 or 1, through a signal channelconsisting of several stages, where transmission through each stage is subject to a fixedprobability of error α. Suppose that X0 = 0 is the signal that is sent and let Xn, be thesignal that is received at the nth stage. Assume that {Xn} is a Markov chain with transitionprobabilities, Poo = P11 = 1- α and P01 = P10 = α, where 0 < α < 1.(a) Determine P {Xo = 0, X1 = 0, X2 = 0}, the probability that no error α occurs up tostage n = 2.(b) Determine the probability that a correct signal is received at stage 2.
8 There are two machines. Machine i operates for an exponential time with rate λi and then fails, its repair time is exponential with rate µi , i = 1, 2. The machines act independently of each other. Define a four-state continuous time Markov chain that jointly describes the condition of the two machines. Use the assumed independence to compute the transition rates for this chain
A manufacturer has a machine that if it ran all day today has a proba- bility of 0.2 of breaking down sometime during the day tomorrow. When the machine breaks down, it goes offline for the remainder of the day and then a technician will spend the next day (after the breakdown) repairing it. A newly repaired machine only has a proability of 0.1 of breaking down sometime tomorrow. (a) Formulate the evolution of the status of the machine at the end of the day as a Markov Chain by identifying the three possible states at the end of the day and providing the transition probabilities between these states. (b) Determine the expected first passage times, µij , for all combinations of states i and j where i cannot equal to j (i.e., you don’t need to determine the recurrence times). You must provide the set of equations used to calculate these µij (micro with lowercase i and j). (c) Using your results from (b): (i) Identify the expected number of full days that the machine will remain…
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For the attached transition probability matrix for a Markov chain. a) Compute P{X2 = 3|X0 = 3}.b) Compute limn→∞ P{Xn = 2|X0 = 4}.(Let x denote this probability, and condition on the first transition when the Markov chain begins in state 4.)
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