<s>[INST] conditional expectation of squared standard normal Let $A,B$ independent standard normals. What is $E(A^2|A+B)$?
Is the following ok?
$A,B$ iid and hence $(A^2,A+B),(B^2,A+B)$ iid.
Therefore we have $\int_M A^2 dP = \int_M B^2 dP$ for every $A+B$-measurable set $M$ and hence $E(A^2|A+B) = E(B^2|A+B)$.
We obtain $2 \cdot E(A^2|A+B) = E(A^2|A+B) + E(B^2|A+B) = E(A^2+B^2|A+B) = A^2+B^2$ where the last equation holds since $A^2+B^2$ is $A+B$-measurable.
Finally we have $E(A^2|A+B) = \frac{A^2+B^2}{2}$.
 [/INST] if $A$ and $B$ are independent, then $A^2$ and $A+B$ are not independent because $A^2$ and $A$ are correlated.
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