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

Question

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

Saige Barton · Accepted Answer

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&#039;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  )  )  =  0The 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)

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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