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2. The day-to-day changes in weather for a certain part of the country form a Markov process. Each day is sunny, cloudy, or rainy. • If it is sunny one day, there is a 70% chance that it will be sunny the following day, a 20% chance it will be cloudy, and a 10% chance of rain. • If it is cloudy one day, there is a 30% chance it will be sunny the following day, a 50% chance it will be cloudy, and a 20% chance of rain. • If it rains one day, there is a 60% chance that it will be sunny the following day, a 20% chance that it will be cloudy and a 20% chance of rain. (a) Create a transition diagram that describes this scenario. (b) Create a stochastic matrix that describes this scenario. Is this scenario ergodic or absorbing? Explain.

2. The day-to-day changes in weather for a certain part of the country form a Markov process. Each day is sunny, cloudy, or rainy. • If it is sunny one day, there is a 70% chance that it will be sunny the following day, a 20% chance it will be cloudy, and a 10% chance of rain. • If it is cloudy one day, there is a 30% chance it will be sunny the following day, a 50% chance it will be cloudy, and a 20% chance of rain. • If it rains one day, there is a 60% chance that it will be sunny the following day, a 20% chance that it will be cloudy and a 20% chance of rain. (a) Create a transition diagram that describes this scenario. (b) Create a stochastic matrix that describes this scenario. Is this scenario ergodic or absorbing? Explain.

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 9EQ
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2. The day-to-day changes in weather for a certain part of the country form a Markov process. Each day is sunny, cloudy, or rainy. • If it is sunny one day, there is a 70% chance that it will be sunny the following day, a 20% chance it will be cloudy, and a 10% chance of rain. • If it is cloudy one day, there is a 30% chance it will be sunny the following day, a 50% chance it will be cloudy, and a 20% chance of rain. • If it rains one day, there is a 60% chance that it will be sunny the following day, a 20% chance that it will be cloudy and a 20% chance of rain. (a) Create a transition diagram that describes this scenario. (b) Create a stochastic matrix that describes this scenario. Is this scenario ergodic or absorbing? Explain. (c) Suppose that today, there is a 42% chance of sun, 38% chance of clouds, and 20% chance of rain. Using matrix multiplication, predict the weather tomorrow, next Thursday (in 7 days), and in two weeks (in 14 days). (d) Find the eigenvalues and eigenvectors for this transition matrix. (e) In the long run, what percentage of days will be sunny? Cloudy? Rainy? Explain.
Transcribed Image Text:2. The day-to-day changes in weather for a certain part of the country form a Markov process. Each day is sunny, cloudy, or rainy. • If it is sunny one day, there is a 70% chance that it will be sunny the following day, a 20% chance it will be cloudy, and a 10% chance of rain. • If it is cloudy one day, there is a 30% chance it will be sunny the following day, a 50% chance it will be cloudy, and a 20% chance of rain. • If it rains one day, there is a 60% chance that it will be sunny the following day, a 20% chance that it will be cloudy and a 20% chance of rain. (a) Create a transition diagram that describes this scenario. (b) Create a stochastic matrix that describes this scenario. Is this scenario ergodic or absorbing? Explain. (c) Suppose that today, there is a 42% chance of sun, 38% chance of clouds, and 20% chance of rain. Using matrix multiplication, predict the weather tomorrow, next Thursday (in 7 days), and in two weeks (in 14 days). (d) Find the eigenvalues and eigenvectors for this transition matrix. (e) In the long run, what percentage of days will be sunny? Cloudy? Rainy? Explain.
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