Transcription of Problems in Markov chains - ku
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Problems inMarkov chainsDepartment of Mathematical SciencesUniversity of CopenhagenApril 2008 This collection of Problems was compiled for the course Statistik 1B. It con-tains the Problems in Martin Jacobsen and Niels Keiding: Markovk der(KUIMS 1990), and 1990 Torben MartinussenJan W. NielsenJesper MadsenIn this edition a few misprints have been correctedSeptember 1992S ren Feodor NielsenIn this edition a few misprints have been correctedSeptember 1993 Anders BrixTranslated into English. A number of additional Problems have been 2007 Merete Grove JacobsenS ren Feodor NielsenMisprints corrected and additional Problems 2008S ren Feodor Nielsen21. Conditional independenceProblem that there are functions (of sets)fzandgzsuch thatfor all setsAandBwe haveP{X A, Y B|Z=z}=fz(A)gz(B)for everyz. Show thatXandYare conditionally independent the result of the previous problem to show, or showdirectly, that ifP{X=x, Y=y, Z=z}=f(x, z) g(y, z)for some functionsfandgthenXandYare conditionally independent thatXandYare conditionally independent givenZifand only ifP{X A|Y=y, Z=z}=P{X A|Z=z}for every (measurable) setAand ((Y, Z)(P)-almost) every (y, z).
The theory for these processes can be handled within the theory for Markov chains by the following con-struction: Let Yn = ... urn and replacing it with c+1 marbles of the same colour, c ∈ N. Define the stochastic process {Xn}n≥1 by Xn = (1 if the n’th marble drawn is black
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