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Unformatted text preview: ( Z = 1) + 2 Â· P ( Z = 2) + 4 Â· P ( Z = 4) + 8 Â· P ( Z = 8) + 16 Â· P ( z = 16) = 14 3 = E ( X ) Â· E ( Y ) E ( XY ) = E ( Z ) On the next page, we mimic the proof of Lemma 5.28, using these X,Y . Reading the proof with this example in mind, might make the proof more understandable. E ( X ) E ( Y ) = X x âˆˆ{ 1 , 2 , 4 } xP ( X = x ) X y âˆˆ{ 1 , 2 , 4 } yP ( Y = y ) = X x âˆˆ{ 1 , 2 , 4 } X y âˆˆ{ 1 , 2 , 4 } xyP ( X = x ) P ( Y = y ) = X z âˆˆ{ 1 , 2 , 4 , 8 , 16 } z X x,y âˆˆ{ 1 , 2 , 4 } xy = z P ( X = x ) P ( Y = y ) = X z âˆˆ{ 1 , 2 , 4 , 8 , 16 } z X x,y âˆˆ{ 1 , 2 , 4 } xy = z P ` ( X = x ) âˆ§ ( Y = y ) Â´ = X z âˆˆ{ 1 , 2 , 4 , 8 , 16 } zP ( Z = z ) = E ( Z ) = E ( XY ) Ind of X,Y...
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 Spring '10
 M.J.Golin
 Computer Science, Probability theory, yp, independent random variables, xy =z, diï¬€erent probability weights

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