In my statistical signal modelling class we were given the following task:
For simplicity, let's assume $E(y) = E(x) = 0$. Show that if the real variables $x$ and $y$ are normally distributed, then the equation $E(y|x) = \operatorname{corr}(y,x) \cdot x/\operatorname{var}(x)$ holds.
We were also given a hint: write the assisting variable $s = y - \operatorname{corr}(x,y)\cdot x/\operatorname{var}(x)$ and show that it's uncorrelated with $x$, meaning that $E(sx) = E(s)E(x) = 0$. Because $s$ and $x$ together are normally distributed, it follows that $s$ and $x$ are uncorrelated, ie. the conditional probability of $s$ doesn't depend on $x$. Now you can easily write $y$ using $s$, and calculate $E(y|x)$.
I've been hacking at this for 3 hours without much progress so any help is appreciated