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1. What are the transition probabilities poi,p11, and p22 in this Markov chain? P01 = P1 = P22= Notation: In all of the following questions, let X; denote the number of messages remaining in the mailbox after Bob checks and answers messages on day i, e.g. X₂ = 1 indicates that 1 message is left in the mailbox after Bob checks on day 2. 2. What is the probability P(X₂ = 1, X3 = 2, X₁ = 0|X₁ = 0) (i.e. the probability that X₂ = 1, X3 =2 AND X₁ = 0 given that X₁ = 0)? P(X₂ = 1, X3 = 2, X₁=0|X₁ = 0) = 3. Given that X₁ = 0, what is the probability that in each of the following three days there is never 2 messages remaining after he checks the mailbox, i.e. X; #2 for all i = 2, 3, 4? 4. Given that X₁ = 0, what is probability that X₂=0? Note that there are three transitions in the process.

1. What are the transition probabilities poi,p11, and p22 in this Markov chain? P01 = P1 = P22= Notation: In all of the following questions, let X; denote the number of messages remaining in the mailbox after Bob checks and answers messages on day i, e.g. X₂ = 1 indicates that 1 message is left in the mailbox after Bob checks on day 2. 2. What is the probability P(X₂ = 1, X3 = 2, X₁ = 0|X₁ = 0) (i.e. the probability that X₂ = 1, X3 =2 AND X₁ = 0 given that X₁ = 0)? P(X₂ = 1, X3 = 2, X₁=0|X₁ = 0) = 3. Given that X₁ = 0, what is the probability that in each of the following three days there is never 2 messages remaining after he checks the mailbox, i.e. X; #2 for all i = 2, 3, 4? 4. Given that X₁ = 0, what is probability that X₂=0? Note that there are three transitions in the process.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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1. What are the transition probabilities po1,P11, and p22 in this Markov chain? Poi = Pii = P22 = Notation: In all of the following questions, let X; denote the number of messages remaining in the mailbox after Bob checks and answers messages on day i, e.g. X, = 1 indicates that 1 message is left in the mailbox after Bob checks on day 2. 2. What is the probability P(X, = 1, X3 = 2, X4 = 0|X1 = 0) (i.e. the probability that X2 = 1, X3 = 2 AND X4 = 0 given that X1 = 0)? P(X2 = 1, X3 = 2, X4 = 0|X1 = 0) = 3. Given that X1 = 0, what is the probability that in each of the following three days there is never 2 messages remaining after he checks the mailbox, i.e. X; + 2 for all i= 2,3, 4? 4. Given that X1 = 0, what is probability that X4 = 0? Note that there are three transitions in the process.
Transcribed Image Text:1. What are the transition probabilities po1,P11, and p22 in this Markov chain? Poi = Pii = P22 = Notation: In all of the following questions, let X; denote the number of messages remaining in the mailbox after Bob checks and answers messages on day i, e.g. X, = 1 indicates that 1 message is left in the mailbox after Bob checks on day 2. 2. What is the probability P(X, = 1, X3 = 2, X4 = 0|X1 = 0) (i.e. the probability that X2 = 1, X3 = 2 AND X4 = 0 given that X1 = 0)? P(X2 = 1, X3 = 2, X4 = 0|X1 = 0) = 3. Given that X1 = 0, what is the probability that in each of the following three days there is never 2 messages remaining after he checks the mailbox, i.e. X; + 2 for all i= 2,3, 4? 4. Given that X1 = 0, what is probability that X4 = 0? Note that there are three transitions in the process.
Bob's phone plan has a voicemail service with the storage capacity of maximum 2 voice messages. Each morning, Bob checks and answers the voice messages according to his available time during that morning. The number of messages that Bob answer each morning is 0 with probability po = 0.2, 1 with probability p = 0.3, 2 with probability p2 = 0.5. After Bob answers a message, the message is removed from the voice mailbox. The number of messages that Bob receives the day before is 0 with probability qo = 0.1, 1 with probability q = 0.5, and 2 with probability q2 = 0.4. If the voice mailbox is already full, all newly arrived messages will be discarded. For simplicity, assume that there is no voice messages received in the mornings during the time he checks. A Markov chain model: The above system can be modeled by a discrete-time homogeneous Markov chain as in the figure below: Poo P11 P22 Poi P12 1 2 P10 P21 Po2 P20 In this Markov chain, we have three states V = {0, 1, 2} indicating the number of messages remaining in Bob's voicemail box after he checks his voicemails in the mornings. We derive that poo = 0.7.
Transcribed Image Text:Bob's phone plan has a voicemail service with the storage capacity of maximum 2 voice messages. Each morning, Bob checks and answers the voice messages according to his available time during that morning. The number of messages that Bob answer each morning is 0 with probability po = 0.2, 1 with probability p = 0.3, 2 with probability p2 = 0.5. After Bob answers a message, the message is removed from the voice mailbox. The number of messages that Bob receives the day before is 0 with probability qo = 0.1, 1 with probability q = 0.5, and 2 with probability q2 = 0.4. If the voice mailbox is already full, all newly arrived messages will be discarded. For simplicity, assume that there is no voice messages received in the mornings during the time he checks. A Markov chain model: The above system can be modeled by a discrete-time homogeneous Markov chain as in the figure below: Poo P11 P22 Poi P12 1 2 P10 P21 Po2 P20 In this Markov chain, we have three states V = {0, 1, 2} indicating the number of messages remaining in Bob's voicemail box after he checks his voicemails in the mornings. We derive that poo = 0.7.
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