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3 years ago

ECE 520.651 Random Signal Analysis

ECE 520.651 Random Signal Analysis

Extra work-space

Extra work-space 12

Question Nō 1: Sample Spaces, Event Spaces and Random Variables.(1a) Show that for any mapping X : Ω → IR, inverse images under X preserve the followingset operations.(12 points)(i) Complementation: if A ⊂ IR, then X −1 (A c ) = [X −1 (A)] c .(ii) Countable Unions: if A 1 , A 2 , . . . ⊂ IR, then X −1 ( ⋃ ∞i=1 A i ) = ⋃ ∞i=1 X −1 (A i ).(iii) Countable Intersections: if A 1 , A 2 , . . . ⊂ IR, then X −1 ( ⋂ ∞i=1 A i ) = ⋂ ∞i=1 X −1 (A i ).3

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