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A3. The weather in Markova is either sunny, cloudy or raining, and the weather condi- tions on successive mornings follow a Markov chain with transition matrix Sunny Cloudy Raining Sunny 0.7 0.2 0.1 Cloudy 0.2 0.6 0.2 Raining 0.1 0.4 0.5 (a) If it is raining on Friday, what is the probability of a sunny weekend? (b) If it is sunny on Monday, what is the probability that it is sunny on Wednesday? (c) If it was sunny on Monday and Wednesday, what is the probability that it rained on Tuesday? (d) Find the stationary probabilities for this Markov chain and hence determine the proportion of sunny days in Markovia. (8

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Than sihon rlatrix is given Sunny Cloudy Raunumy unny 0.7 0.2 0.2 O. 6 o.2 Ratu 0.1 o.1 0.5 a Gven at twen hat it is Ramun on Friday it is Ran m friday we ued aming on fuiday, We nue 2 to find the praba bility of a Sunny Wee kend Yob a biln unm Sum l.e P (Sunda, Sunn섭 / Sahunda Sunny) o, o구qiwen hat it is sunny on nued to fund re pro ba bihty thatwe on Wednesdar b) ond IS hWednes P Wed S uuny MonS t P (Wed = Sunny / Tui -cloud Sunny Tue Ra 0.Y9 0.04 O.01 d nesday, C Giem that itwa Sunny on Honday and tha probability hat it haumd ned to fund Hhu Tuuday we on Sinice thu weathu n Tusday dpends only on tha uwtattuer of pravreuu day eoday (te aue Jwen poc s a Maxko Chain) we olo not ued mfton mahion for Hed rea day hed nes day Sunny Wede Sury HomSum Sunry 0.1d) A startomary dis bvibotion of a Htar ko is a probability dlistribotión that emischog in thu Harkov Chain as time progresses gt es gwen by a vec for TT. ner, Trenu ú trof thnee elements say T, , ,and T, (for re tram sihon matin P) tu statu,haoy トHobs, should equal T o, th Sum of Ftem 1123-4찌 = 1.2 170 TT3 = 0.2-0.1x6 24-6 140 170 1-_60-23-1-88 = (40 170 , _ 32. Hence, T228 Ito Go Ito Plus ady) = 28 D. I64 p( Raining) 82 0.483 35%, ot days au Sunny m Markovi a

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