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Problems in Markov chains - ku

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.

for every (measurable) set A and ((Y,Z)(P)-almost) every (y,z). Thus if X and Y are conditionally independent given Z, then X is inde-pendent of Y given Z. Problem 1.4 Suppose that X, Y and Z are independent random variables. Show that (a) X and Y are conditionally independent given Z (b) X and X +Y +Z are conditionally independent given X +Y

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