Show that, for random variables X and Z, E[(X-Y)^(2)]=E[X^(2)]-E[Y^(2)] where Y=E[X|Z]

frobirrimupyx 2022-09-11 Answered
Show that, for random variables X and Z, E [ ( X Y ) 2 ] = E [ X 2 ] E [ Y 2 ] where Y=E[X|Z]
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Answers (1)

Saige Barton
Answered 2022-09-12 Author has 15 answers
Write out the left side. We have that
E ( ( X Y ) 2 ) = E ( X 2 + Y 2 2 X Y )
= E ( X 2 ) + E ( Y 2 ) 2 E ( X Y )
Now, what is E(XY)? Let's prove that it is equal to E ( Y 2 ).
E ( X Y Y 2 ) = E ( Y ( X Y ) ) = E ( E ( X | Z ) ) ( X E ( X | Z ) ) = 0
The expression above is equal to zero, since we know that the projection E(X|Z) and the connection between the point X and the projection E(X|Z) are orthogonal. Therefore E ( X Y ) = E ( Y 2 ) which implies
E ( ( X Y ) 2 ) = E ( X ) E ( Y 2 )
Result:
Use the fact that X-E(X|Z) is orthogonal to E(X|Z)

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