<s>[INST] Conditional Expectation of E(2X+Y|X-Y=1) I met these two problems. The first question reads: what is $E(X|X+Y=1)$ 
given that $X$ and $Y$ are both independent standard normal random variables. 
The second reads that $X$ and $Y$ are correlated standard normal random variables at a correlation of $\rho$, then what is $E(2X+Y|X-Y=1)?$
I tried to first find the sample space of given condition that $X+Y=1$ and $X-Y=1$. I realized that they are both sample space with $0$ probability since the two space are reduced to $\rm I\!R^1$ from $\rm I\!R^2$, therefore the conditions should have $0$ probability. In that case
$E(X|X+Y=1) = E(1-Y) = 1-0\quad$ since $\quad Y\subset    \rm I\!R^1$ 
$E(2X+Y|X-Y=1) = E(2X+X-1) = E(3X-1) = -1$ since $\quad X\subset    \rm I\!R^1$ 
are these answers correct?
 [/INST] We assume that $X$ and $Y$ are jointly normal. Then $X-Y$ and $X+Y$ are independent. Moreover,
\begin{align*}
E\big(X \mid X+Y=1 \big) &= \frac{1}{2}E\big( (X+Y)+(X-Y) \mid X+Y=1\big)=\frac{1}{2},
\end{align*}
and
\begin{align*}
E\big(2X+Y \mid X-Y=1 \big) &=E\big(X \mid X-Y=1 \big) +E\big(X+Y \mid X-Y=1 \big)\\
&=\frac{1}{2} + E(X+Y) = \frac{1}{2}.
\end{align*}
</s>