Tutorial 5.pdf - THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT3910 Finanical Economics I 2019-2020 Semester 1 Tutorial

Tutorial 5.pdf - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT3910 Finanical Economics I 2019-2020 Semester 1 Tutorial 5 1 Summary Conditional Expectation In the probability space ( , P ), denote the power set of by F 0 , and we have two sigma fields G F 0 , F F 0 . If X F , A G , then the conditional expectation on an event A can be calculated by E [ X | A ] = ஀? i x i P ( { ω : X ( ω ) = x i } A ) P { A } . A special case is that G F . It is safe to ignore the intersections that are empty, that is, the set of events { B i A : B i F for i = 1 , 2 , . . . , m } is considered. Thus E [ X | A ] = m ஀? i =1 x i P ( B i ) P ( A ) , where we note that X ( ω ) = x i if ω B i for i = 1 , 2 , . . . , m . For G = σ { A 1 , A 2 , ..., A n } , where n i =1 A i = and A i ’s are disjoint for i = 1 , · · · , n . In other words, { A i } forms the partition of . The conditional expectation (conditional on σ -field G ) can be calculated by E [ X |G ] = ஀? ஀? ஀? ஀? ஀?
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