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Math 312 Lecture Notes Markov Chains - Colgate University

Math312 LectureNotesMarkov ChainsWarrenWeckesserDepartment of MathematicsColgateUniversityUpdated,30 April2005 Markov ChainsA ( nite) Markov chainis a processwitha nitenumber of states(oroutcomes,or events)in whichtheprobability of beingin a particularstateat stepn+ 1 ; S2; : : : ; Srgbe ~pn= (1)be thevectorof probabilitiesof each stateat stepn. Thatis, theith entryof~pnis theprobabilitythattheprocessis in stateSiat stepn. For such a probability vector,p1+p2+ +pr= Prob(Staten+ 1 isSijStatenisSj);(2)andletP=26664p11p12 p1rp21p22 (3)Thatis,pijis the(conditional)probability of beingin stateSiatstepn+ 1 giventhattheprocesswas in stateSjat theMarkov canbe usefulto label therowsandcolumnsofPwiththestates,as in thisexamplewiththreestates:Statenz}|{S1S 2S3 Staten+ 18> <>:S1S2S3264p11p12p13p21p22p23p31p32p33375 1 Thefundamentalproperty of a Markov chainis that~pn+1=P~pn:(4)Givenaninitialprobabil ity vector~p0, we candeterminetheprobability vectorat any stepnbycomputingtheiteratesof a thetransitionmatrixcanalsobe representedin atransitiondiagram.

Math 312 Lecture Notes Markov Chains Warren Weckesser Department of Mathematics Colgate University Updated, 30 April 2005 Markov Chains A ( nite) Markov chain is a process with a nite number of states (or outcomes, or events) in which

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