A certain town never has two sunny days in a row. Each day is classified as being either sunny, cloudy(but dry), or rainy. If it is sunny one day, then it is equally likely to be either cloudy or rainy the next day. If it is rainy or cloudy one day, then there is one chance in two that it will be the same the next day, and if it changes then it is equally likely to be either of the other two possiblilities. In the long run, what proportion of days are sunny? what proportion are cloudy? Show that the markov chain is time reversible.

The proportion of cloudy and rainy days in long run would be 40 % each and sunny days 20%.

The Markov chain is time reversible because of the existence of the above proportions.

Step-by-step explanation:

In the question, we were given a case of Markov chain with 3 states: Sunny(1) , Cloudy(2) and Rainy(3) . According to the given conditions, the transition matrix can be obtained as below:

\begin{pmatrix} 0 &0.5 &0.5 \\ 0.25&0.5 &0.25 \\ 0.25& 0.25 &0.5 \end{pmatrix}

Let us denote the proportion of sunny, cloudy and rainy days in long run as S,C and R

From  column 1, we get:

S = 0.25C + 0.25R(1)

From column 2, we get:

C = 0.5S + 0.5C + 0.25R

0.5C = 0.5S + 0.25R

2C = 2S + R

C = S + 0.5R(2)

Substituting the value of C from (2) in (1) we get:

S = 0.25(S + 0.5R) + 0.25R

S = 0.25S + 0.125R + 0.25R

0.75S = 0.375R

S = 0.5R

Thus,

C = S + 0.5R = R

Knowing that the sum of the 3 proportions should be equal to 1.

That is, S+C+R =1

Putting everything in terms of R we get:

0.5R + R + R = 1

R = 1/2.5 = 0.4

C = 0.4

S = 0.2

Hence, the proportion of cloudy and rainy days in long run would be 40 % each and sunny days 20%.

The Markov chain is time reversible because of the existence of the above proportions.

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