Markov Chain1

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1、Chapter2:MarkovChainsDefinition.Astochasticprocess{X(t),t∈T}isacollectionofX-valuedrandomvariablesindexedbyasetT(“time”).•X(t)isthestateoftheprocessattimet.•IftheindexsetTiscountable,itisadiscrete-timeprocess;otherwiseitisacontinuous-timeprocess.CourtesyofProf.BobL

2、i2.1Somepuzzlesinstochasticprocess2.1.1PatternoccurrenceinrepeatedcointossP{Head}=P{Tail}=½.Inaverylongbinarysequence,allthe64length-6patternsappearequallyfrequently.HTHTTTTHTHHTHTTTHTTTTHTHHTHTTTHTHTHTTHTTHTHTTTHTHHTHTTHTHHTHHTHHTHTTHTHHTHHTHHTHTTTHTHHTTHHTTTHTTTT

3、THTHHTHHTHHTHTTHTHHTHTTTTHTHHTHTTTHTHHTHHTHHTTTHHTHHTHTHTTHTHTHHTHTHTTHTHHHHTHTHTTHTTHTTTHTHTHTHTHHTTTTHHTHTTHTHTHTHTTHHTHHTHTHTHTHHTHTTTHTTTHHTHTHTHHTTHHTHTTHTTHTTHTHHTHHTHHTHTTHTTHTHTHTHTTHTHTHTTHTHTHHTTHTTHTHTHTTHTHTHHHTHHTHTHTTHTTHTHHTHTTHTHHTHTHHTHTHTTHTTHTHTH

4、THTTHTHTHTHTTHTHTTTTHTHHTTHHHTTHHTTTHHHHTHHTHTHHHHHHTTHTHTHTHHTHHTHTTHHTHTHTTHHTHTHTTTHTHTHHTHTHHTTHTHHTTHHTTHHHTTHTTHTTHTTHHTHHTHTTHTTHTTHTHTHHTHTHTTHTTHTHTTHTTHTTHTHHTHTTHTTTHTHTTHTHHTTH…So,theaveragewaitingtimeforeverylength-6patternshouldbe64.Right?WilliamFelle

5、r’sbiblicaltextbookThebookhasfirstraisedthesequestions:•AveragewaitingtimeforHHHHHH=64?•AveragewaitingtimeforHHTTHH=64?WilliamFeller’sbiblicaltextbookThroughlongderivationbymarkovchain,•AveragewaitingtimeforHHHHHH=126•AveragewaitingtimeforHHTTHH=70WilliamFellerfoun

6、dthesenumberscounter-intuitive.ParadoxesinpatternoccurrenceHTHTTTTHTHHTHTTTHTTTTHTHHTHTTTHTHTHTTHTTHTHTTTHTHHTHTTHTHHTHHTHHTHTTHTHHTHHTHHTHTTTHTHHTTHHTTTHTTTTTHTHHTHHTHHTHTTHTHHTHTTTTHTHHTHTTTHTHHTHHTHHTTTHHTHHTHTHTTHTHTHHTHTHTTHTHHHHTHTHTTHTTHTTTHTHTHTHTHHTTTTHHTH

7、TTHTHTHTHTTHHTHHTHTHTHTHHTHTTTHTTTHHTHTHTHHTTHHTHTTHTTHTTHTHHTHHTHHTHTTHTTHTHTHTHTTHTHTHTTHTHTHHTTHTTHTHTHTTHTHTHHHTHHTHTHTTHTTHTHHTHTTHTHHTHTHHTHTHTTHTTHTHTHTHTTHTHTHTHTTHTHTTTTHTHHTTHHHTTHHTTTHHHHTHHTHTHHHHHHTTHTHTHTHHTHHTHTTHHTHTHTTHHTHTHTTTHTHTHHTHTHHTTHTHHTTHH

8、TTHHHTTHTTHTTHTTHHTHHTHTTHTTHTTHTHTHHTHTHTTHTTHTHTTHTTHTTHTHHTHTTHTTTHTHTTHTHHTTH…Inthelongrun,thepatternHHTTHHoccursequallyfrequentlyasanyotherl