I have a question attached below, and I really do't know how to solve it. Is there anyone can help me with this? $x,y\sim N(0,1)$, independent, what is $E(x\mid x+y=1)$, what about variance? What I am thinking is to calculate $P(x\mid x+y=1)$ first, $P(x\mid x+y=1)=1=P(x+y=1\mid x)\cdot P(x)/P(x+y=1)$, but I don't know how to express it...
As is the case as soon as $x$ and $y$ are i.i.d., the symmetry $(x,y)\to(y,x)$ shows that $\mathbb E(x\mid x+y)=\mathbb E(y\mid x+y)=\frac12\mathbb E(x+y\mid x+y)=\frac12(x+y)$. In particular, $\mathbb E(x\mid x+y=1)=\frac12$.