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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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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