Conditional expectation of two correlated RVs $X$ and $Y$ are two correlated random variables. I am trying to estimate $E(X\mid Y)$ given $E(X)$, $E(Y)$, $\rho(X,Y)$, $\sigma(X)$ and $\sigma(Y)$. Could someone point me how to go about it. What if $X$ and $Y$ are vectors of RVs with $\Sigma_X$ and $\Sigma_Y$ being their covariance matrices.

Thinking a bit about it (for non-vector case) what's the problem with following approach:
$\hat{\beta}$ in OLS $\hat{X} = \hat{\beta}_0 + \hat{\beta}_1 Y_1 + \cdots + \hat{\beta}_n Y_n $ is estimated as follows:
$$ \hat{\beta}_i = \frac{ Cov(Y_i, X) }{\sigma^2(Y_i) }$$
$$ \hat{\beta}_0 = E[X]$$
Above OLS equation can be re-written as:
$$E[X | Y_1, ..., Y_n] = E[X] + \Sigma_i\frac{ Cov(Y_i, X) }{\sigma^2(Y_i)}E[Y_i]$$
In case $n=1$ and say $Y_1 = Y$:
$$E[X | Y] = E[X] + \frac{ Cov(Y, X) }{\sigma^2(Y)}E[Y]$$
$$E[X | Y] = E[X] + \frac{ \rho(Y, X) \sigma(X) \sigma(Y) }{\sigma^2(Y)}E[Y]$$
$$E[X | Y] = E[X] + \frac{ \rho(Y, X) \sigma(X)  }{\sigma(Y)}E[Y]$$
