exercise_on_pro.ess_chp3-lastcbj

# exercise_on_pro.ess_chp3-lastcbj - Printed April 16, 2009...

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Printed April 16, 2009 Exercise on Probability Essentials By Author. Jean Jacod, Philip Protter Byoung jin Choi e-mail: choibj@chungbuk.ac.kr Abstract In this book we solve a exercise of the Probability Essentials wrote by Jean Jacod, Philip Protter. Author thanks to graduate student of Department of Mathematics in Chungbuk National University in Republic of Korea. 1

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1 Introduction 2 Axioms of probability 3 Conditional Probability and Independence Exercise 3.2 Suppose P ( C ) > 0 and A 1 , A 2 , ··· , A n are all pairwise disjoint. Show that P ( n n =1 A i | C ) = n X n =1 P ( A i | C ) . Proof. Since P ( n n =1 A i | C ) = P ( n n =1 A i C ) P ( C ) and P ( n n =1 A i C ) = P ( n n =1 ( A i C )) = n X n =1 P ( A i C ) . Hence this proof is completed. Exercise 3.6 Let ( A n ) n 1 ∈ A and ( B n ) n 1 ∈ A and A n A and B n B with P ( B ) > 0 and P ( B n ) > 0, all n . Show that (a) lim n →∞ P ( A n | B ) = P ( A | B ) (b) lim
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## exercise_on_pro.ess_chp3-lastcbj - Printed April 16, 2009...

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