Double Expectation Theorem

Double Expectation Theorem - x x X y P ( X = x,Y = y ) = X...

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Double Expectation Theorem: E ( X ) = E Y [ E X | Y ( X | Y )] = E [ E ( X | Y )] . Proof: (the discrete case) RHS = X y E ( X | Y = y ) P ( Y = y ) = X y X x xP ( X = x | Y = y ) P ( Y = y ) = X y X x x P ( X = x,Y = y ) P ( Y = y ) P ( Y = y ) = X y X x xP ( X = x,Y = y ) = X
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Unformatted text preview: x x X y P ( X = x,Y = y ) = X x xP ( X = x ) = E ( X ) = LHS 2 1...
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