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.
 A: I do not believe this is possible without more information regarding the relationship between the two variables.
For example, let's say that if X is positive, Y is negative, and vice-versa, and independently, they are two normally-distributed variables with a mean of $0$ and variance of $1$. You sample both of them, and then if their signs are the same, then flip one of them, e.g., by always choosing Y.
The correlation of these two variables will be negative, about $-0.63$. However, the only information you get from knowing X or Y is the sign. The expected value in either case would be roughly $0.8$ or $-0.8$.
If you have two normally-distributed variables that can be described using a standard correlation matrix $\Sigma_{xy}$, then you will have a unique value of $E(X|Y)$ for every value of $Y$.
There is definitely a more robust way of demonstrating more general cases of how this problem needs more constraints, but this is probably the easiest to visualize.
Here's a plot showing how two normal, correlated distributions can have a vastly different joint PDF, even if you know the correlation, their expected values, and standard deviations.

# code to generate plot
X = rnorm(10000)
Y = rnorm(10000)

Y = -abs(Y) * sign(X)

plot(X,Y,main='Signs are opposite')
cor(X,Y)


XYc = MASS::mvrnorm(
  10000,
  mu=c(0,0),
  Sigma=matrix(c(1,cor(X,Y),cor(X,Y),1), nrow=2)
)

par(mfcol=c(1,2))
plot(X,Y,main='Opposite-signed X and Y',pch='.')
plot(XYc, xlab='X',ylab='Y', main='Correlated X and Y',pch='.')
```

A: 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 $Y_1 = Y$:
$$E[X | Y] = E[X] + \frac{ Cov(Y, X) }{\sigma^2(Y)}E[Y]$$
$$E[X | Y] = E[X] + \frac{ \rho(Y, X) \sigma(X) \sigma(Y) }{\sigma^2(Y)}E[Y]$$
$$E[X | Y] = E[X] + \frac{ \rho(Y, X) \sigma(X)  }{\sigma(Y)}E[Y]$$
