Let $X \sim N(0, 1)$ and $Y \sim N(0, 1)$ and $\mathbb E[XY]=\rho$. Can one say anything about the conditional expectation $\mathbb E[X \mid Y]$?
In general, this clearly does not seem to work, because $\mathbb E[X \mid Y]$ will depend on the joint distribution of $X, Y$ while $\rho$ only measure linear relationship between them. But is it possible to say anything in the Normal case?
It is mentioned in this wikipedia article that if $X, Y$ have a jointly Normal distribution then $\rho=0$ implies independence, i.e. we know $E[X \mid Y] = E[X]$. But is this as far as one can go in terms of inferences about $E[X \mid Y]$ from the correlation?
If we are told that $X$ and $Y$ are jointly Normal, then we know that
$$E[X|Y] = E[X] + \rho \frac{\sigma_X}{\sigma_Y}( Y-E[Y])$$
which in your case reduces to $$E[X|Y] = \rho Y$$
In general, if we only know that the variables are marginally normal, then I don't think there's much to say. Calling $E[X|Y]=g(Y)$ we know that $$E[XY] = E[E[XY|Y]] =E[Y g(Y)] )= \int y \, g(y) \exp{(-y^2/2}) \,dy =\rho $$ and $$E[X]=E[E[X|Y]]=E[g(Y)]=\int g(y) \exp{(-y^2/2}) \,dy=0$$ but this does not say much.