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Problem 2. Tlaloc has 4 umbrellas, each either t home or at work. Each time he goes to work or back, it rains with probability p, independently of all other times. If it rains and there is at least one umbrella with him, he takes it. Otherwise, he gets wet. Let Xn be the number of umbrellas at his current location after n trips (so n even corresponds to home and n odd to work. (a) Find the transition probabilities for the Markov chain Xn. (b) Show that it is reversible and find the stationary distribution. (c) What is the probability that Tlaloc gets wet on any given trip? (d) What value of p maximizes this probability? (e) What happens if he has N umbrellas for N other than 4?

Problem 2. Tlaloc has 4 umbrellas, each either t home or at work. Each time he goes to work or back, it rains with probability p, independently of all other times. If it rains and there is at least one umbrella with him, he takes it. Otherwise, he gets wet. Let Xn be the number of umbrellas at his current location after n trips (so n even corresponds to home and n odd to work. (a) Find the transition probabilities for the Markov chain Xn. (b) Show that it is reversible and find the stationary distribution. (c) What is the probability that Tlaloc gets wet on any given trip? (d) What value of p maximizes this probability? (e) What happens if he has N umbrellas for N other than 4?

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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Problem 2. Tlaloc has 4 umbrellas, each either t home or at work. Each time he goes to work or back, it rains with probability p, independently of all other times. If it rains and there is at least one umbrella with him, he takes it. Otherwise, he gets wet. Let Xn be the number of umbrellas at his current location after n trips (so n even corresponds to home and n odd to work. (a) Find the transition probabilities for the Markov chain Xn. (b) Show that it is reversible and find the stationary distribution. (c) What is the probability that Tlaloc gets wet on any given trip? (d) What value of p maximizes this probability? (e) What happens if he has N umbrellas for N other than 4?
Transcribed Image Text:Problem 2. Tlaloc has 4 umbrellas, each either t home or at work. Each time he goes to work or back, it rains with probability p, independently of all other times. If it rains and there is at least one umbrella with him, he takes it. Otherwise, he gets wet. Let Xn be the number of umbrellas at his current location after n trips (so n even corresponds to home and n odd to work. (a) Find the transition probabilities for the Markov chain Xn. (b) Show that it is reversible and find the stationary distribution. (c) What is the probability that Tlaloc gets wet on any given trip? (d) What value of p maximizes this probability? (e) What happens if he has N umbrellas for N other than 4?
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