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Suppose I have the followin situation :

The weather can either be "bad" or "good".

If the weather yesterday was "good" and the weather today is "good", then the probability that the weather will be "good" tomorrow is $p_1$

If the weather yesterday was "good" and the weather today is "bad", then the probability that the weather will be "good" tomorrow is $p_2$

If the weather yesterday was "bad" and the weather today is "good, then the probability that the weather will be "good" tomorrow is $p_3$

If the weather yesterday was "bad" and the weather today is "bad", then the probability that the weather will be "good" tomorrow is $p_4$

(The probability that the weather will be bad tomorrow is 1-$p_{...}$)

How can I represent it as a Markov chain and write a "useful" transition matrix (by useful, I mean such that I would be able to find the stationary vector (i.e. proportion of time spent in each "weather") ) ? I am looking for a general method. I did not found find much about it on the Web...

Thanks for help

MysteryGuy
  • 641
  • Let $0$ indicate "bad" and $1$ "good". The state space is ${0,1}^2$; ordered lexicographically, the transition matrix is $$ P = \pmatrix{1-p_4&p_4&0&0\ 0&0&1-p_3&p_3\1-p_2&p_2&0&0\0&0&1-p_1&p_1 } $$ (That is, if I am interpreting your question correctly.) – Math1000 May 25 '18 at 17:39
  • @Math1000 So how can I do to compute the stationary state from that transition matrix (i.e. know the number of days of good weather) ? Can I compute the eigen vector of P associated to eigen value 1 and sum the last two entries (normalized) ? – MysteryGuy May 25 '18 at 17:52

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