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Markov chain It is known that in rainy seasons, if yesterday and today it rained the...

Markov chain

It is known that in rainy seasons, if yesterday and today it rained the probability that it will rain tomorrow is 0.8, if yesterday it did not rain and today the probability that it will rain tomorrow is 0.5, if yesterday it rained but not today, the probability of that it will rain tomorrow is 0.3, and if it did not rain yesterday and today the probability that it will rain tomorrow is 0.1.

(a) Construct the digraph and the transition matrix for the Markov chain that models this situation.

(b) Suppose it rained yesterday, but not today. Using diagonalization, find the steady state distribution and interpret it

Math   Statistics-And-Probability   
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Answer #1

Answer:

We can transform the above model into a Markov chain by saying that the state at any time is determined by the weather conditions during both that day and the previous day. In other words,

we can say that the process is in:

State 0 if it rained both today and yesterday.

State 1 if it rained today and but not yesterday. State 2 if it rained yesterday but not today.

State 3 if it did not rain either yesterday or today. The preceding would then represent a four-state Markov chain having the following transition probability matrix:

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