The weather on any given day in a particular city can be sunny, cloudy, or rainy. It has been observed to be predictable largely on the basis of the weather on the previous day. Specfically: • if it is sunny on one day, it will be sunny the next day 1/5 of the time, and be cloudy the next day 2/5 of the time • if it is cloudy on one day, it will be sunny the next day 1/10 of the time, and be cloudy the next day 4/5 of the time • if it is rainy on one day, it will be sunny the next day 2/5 of the time, and be cloudy the next day 1/2 of the time Using 'sunny', 'cloudy', and 'rainy' (in that order) as the states in a system, set up the transition matrix for a Markov chain to describe this system. Find the proportion of days that have each type of weather in the long run. 000 P=000 000 Sunny 0 Proportion of days that are Cloudy = 0 Rainy 0
The weather on any given day in a particular city can be sunny, cloudy, or rainy. It has been observed to be predictable largely on the basis of the weather on the previous day. Specfically: • if it is sunny on one day, it will be sunny the next day 1/5 of the time, and be cloudy the next day 2/5 of the time • if it is cloudy on one day, it will be sunny the next day 1/10 of the time, and be cloudy the next day 4/5 of the time • if it is rainy on one day, it will be sunny the next day 2/5 of the time, and be cloudy the next day 1/2 of the time Using 'sunny', 'cloudy', and 'rainy' (in that order) as the states in a system, set up the transition matrix for a Markov chain to describe this system. Find the proportion of days that have each type of weather in the long run. 000 P=000 000 Sunny 0 Proportion of days that are Cloudy = 0 Rainy 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The weather on any given day in a particular city can be sunny, cloudy, or rainy. It has been observed to be predictable largely on the basis of the weather on the previous day. Specifically:
- If it is sunny on one day, it will be sunny the next day 1/5 of the time, and be cloudy the next day 2/5 of the time.
- If it is cloudy on one day, it will be sunny the next day 1/10 of the time, and be cloudy the next day 4/5 of the time.
- If it is rainy on one day, it will be sunny the next day 2/5 of the time, and be cloudy the next day 1/2 of the time.
Using 'sunny', 'cloudy', and 'rainy' (in that order) as the states in a system, set up the transition matrix for a Markov chain to describe this system.
Find the proportion of days that have each type of weather in the long run.
\[
P = \begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\]
\[
\text{Proportion of days that are } \begin{bmatrix}
\text{Sunny} \\
\text{Cloudy} \\
\text{Rainy}
\end{bmatrix} = \begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix}
\]
Explanation:
- The transition matrix \( P \) is intended to represent the probabilities of transitioning from one type of weather to another on the following day.
- The specified matrix is not completed, and the matrix's values should reflect the transition probabilities mentioned above.
- The final vector indicates the long-term proportion of days that are expected to be sunny, cloudy, and rainy. However, it also appears as an incomplete placeholder here.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec2b7ff9-b952-4215-9e0f-f264e2036fb8%2F1a193ec1-5794-4e12-8fc0-ec8f05995bdb%2F0novgmua_processed.png&w=3840&q=75)
Transcribed Image Text:The weather on any given day in a particular city can be sunny, cloudy, or rainy. It has been observed to be predictable largely on the basis of the weather on the previous day. Specifically:
- If it is sunny on one day, it will be sunny the next day 1/5 of the time, and be cloudy the next day 2/5 of the time.
- If it is cloudy on one day, it will be sunny the next day 1/10 of the time, and be cloudy the next day 4/5 of the time.
- If it is rainy on one day, it will be sunny the next day 2/5 of the time, and be cloudy the next day 1/2 of the time.
Using 'sunny', 'cloudy', and 'rainy' (in that order) as the states in a system, set up the transition matrix for a Markov chain to describe this system.
Find the proportion of days that have each type of weather in the long run.
\[
P = \begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\]
\[
\text{Proportion of days that are } \begin{bmatrix}
\text{Sunny} \\
\text{Cloudy} \\
\text{Rainy}
\end{bmatrix} = \begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix}
\]
Explanation:
- The transition matrix \( P \) is intended to represent the probabilities of transitioning from one type of weather to another on the following day.
- The specified matrix is not completed, and the matrix's values should reflect the transition probabilities mentioned above.
- The final vector indicates the long-term proportion of days that are expected to be sunny, cloudy, and rainy. However, it also appears as an incomplete placeholder here.
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