mid09_sol

mid09_sol - STAT 219/ MATH 136 Fall 2009 Midterm Solutions...

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STAT 219/ MATH 136 Fall 2009 Midterm Solutions Friday, October 23 Write your name and sign the Honor code in the blue books provided. This is a closed note and closed book exam. You have 50 minutes to solve all THREE questions, each worth points as marked (maximum of 40). Complete reasoning is required for full credit. You may cite lecture notes and homework sets, as needed, stating precisely the result you use, why and how it applies. Problem 1 (12 total points) Consider the probability space (Ω , F ,P ) where Ω = { 1 , 2 , 3 , 4 , 5 , 6 } , F = 2 Ω and P is the probability measure given by P ( { 1 } ) = 1 4 , P ( { 2 } ) = 1 4 , P ( { 3 } ) = 1 4 , P ( { 4 } ) = 1 12 , P ( { 5 } ) = 1 12 , P ( { 6 } ) = 1 12 . Consider the σ -ﬁeld G ⊂ 2 Ω given by G = {∅ , Ω , { 1 , 2 } , { 3 , 4 , 5 , 6 }} . (a) (4 points) Let A,B,C R and consider Y ( ω ) = AI { 1 , 2 , 3 , 4 } ( ω ) + BI { 3 , 4 } ( ω ) + CI { 1 , 2 , 5 , 6 } ( ω ) . Give some explicit values for A,B,C R so that σ ( Y ) = G . ANS: Any values of A,B,C R such that Y = c 1 I { 1 , 2 } + c 2 I { 3 , 4 , 5 , 6 } for some constants c 1 and c 2 are valid. For example, A = 1, B = 0 and C = 1 gives Y = 2 I { 1 , 2 } + I { 3 , 4 , 5 , 6 } . (b) (3 points) Let X ( ω ) = I { 2 , 3 } . Using your answer from part ( a ), specify σ ( X ) and σ ( X,Y ). ANS: σ ( X ) = { 0 , Ω , { 2 , 3 } , { 1 , 4 , 5 , 6 } . We know that σ ( Y ) = G so that σ ( X,Y ) = σ ( {∅ , Ω , { 1 , 2 } , { 3 , 4 , 5 , 6 } , { 2 , 3 } , { 1 , 4 , 5 , 6 }} ) = {∅ , Ω , { 1 } , { 2 } , { 3 } , { 1 , 3 } , { 1 , 2 } , { 2 , 3 } , { 1 , 2 , 3 } , { 2 , 4 , 5 , 6 } , { 3 , 4 , 5 , 6 } , { 1 , 4 , 5 , 6 } , { 1 , 3 , 4 , 5 , 6 } , { 1 , 2 , 4 , 5 , 6 } , { 2 , 3 , 4 , 5 , 6 }} . (c) (2 points)

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mid09_sol - STAT 219/ MATH 136 Fall 2009 Midterm Solutions...

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