Homework Answers
If today is sunny, probability of being sunny tomorrow is 2/3 and so probability of being cloudy tomorrow is 1/3
If today is cloudy, probability of being cloudy tomorrow is 2/3 and so probability of being sunny tomorrow is 1/3
So our transition matrix P is as follows:
S. C
S. 2/3. 1/3
C. 1/3. 2/3
b) Let X0 be the probability distribution today (S. C) it is (1/2. 1/2)
If we multiply X0 by P, we get probability distribution of Tomorrow that is X1: (1/2*2/3 + 1/2*1/3, 1/2*1/3 + 1/2*2/3) = (0.5, 0.5)
So the required probability of being sunny tomorrow is 0.5
c) We have, X0: (1/2, 1/2)
We multiply this by P to get X1: (1/2, 1/2)
We again multiply X1 by P to get, X2: (1/2, 1/2)
So the probability of being cloudy day after tomorrow is 0.5
d) Let the steady state probabilities be π= πS, πC
We know, πP = π
So we have, 2/3πS + 1/3πC = πS ....1
And 1/3πS + 2/3πC= πC.....2
We also know, πS + πC = 1 .... 3
Solving these equations we get, πS= πC= 1/2
So the steady state equation is (1/2, 1/2)
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