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

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

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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 Solution 1. [ 2014 Final]Let = { 2 , 1 , 1 , 2 } , F = all subsets of , and X ( ω ) = | ω | . Define the probability as follows: P {{ ω }} = 1 4 , for ω = 2 , 1 , 1 , 2 . (a) Find the smallest σ -field w.r.t. which the random variable X is measurable. (b) Let G = σ {{ 2 , 1 } , { 1 , 2 }} . Find E [ X |G ]. Solution: (a) Let F be required smallest σ -field. If X F , then for B B , X 1 ( B ) F is true. From the mapping X ( ω ) = | ω | , we know X ( { 2 , 2 } ) = 2 , X ( { 1 , 1 } ) = 1 , so their inverse images are X 1 ( { 2 } ) = { 2 , 2 } , X 1 ( { 1 } ) = { 1 , 1 } and it is required that X 1 ( { 2 } ) F , X 1 ( { 1 } ) F . Besides, X 1 ( { 2 } ) X 1 ( { 1 } ) = { 2 , 2 } { 1 , 1 } = , which means { 2 , 2 } and { 1 , 1 } are disjoint partitions of . Therefore, F = σ {{ 2 , 2 } , { 1 , 1 }} = { , , { 2 , 2 } , { 1 , 1 }} . (b) Since G is a σ -field generated by disjoint sets { 2 , 1 } and { 1 , 2 } , so E [ X |G ] = ஀? | 2 | · P ( { 2 } { 2 , 1 } ) P ( { 2 , 1 } ) + | 1 | · P ( { 1 } { 2 , 1 } ) P ( { 2 , 1 } ) = 3 2 , ω { 2 , 1 } 1 · P ( { 1 }
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