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Unformatted text preview: equivalent. Proof. Starting with the rst denition, we have X x x P ( X = x ) = X x x X { : X ( )= x } P ( ) . This is because the set ( X = x ) is the union of the disjoint sets { } , where the union is over those for which X ( ) = x . Bringing the x inside the sum, and using the fact that X ( ) = x , our expression is equal to X x X { : X ( )= x } x P ( ) = X x X { : X ( )= x } X ( ) P ( ) . Since the union of { : X ( ) = x } over all possible values of x is just all , we then have X X ( ) P ( ) , which is the second denition. One advantage of the second denition is that linearity is easy. We write E ( X + Y ) = X S ( X ( ) + Y ( )) P ( ) = X [ X ( ) P ( ) + Y ( ) P ( )] = X X ( ) P ( ) + X Y ( ) P ( ) = E X + E Y. 10...
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 Spring '09
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