Estimating conditional mean

Question

Suppose I have a uniform random variable $X$ taking values $\{1,...,n\}$, and two functions $v(X)$ and $w(X)$.

I know that $v(X)$ and $w(X)$ are jointly distributed with correlation $\rho$. (And can assume any distribution structure including joint normal if it makes it easier).

First, I observe all values of $v(X)$ for $x=\{1,...,n\}$ and estimate $E[v(X)]$ as the average of the observations.

My goal is to estimate $E[w(X)]$ without sampling $w(X)$ by using the values of $v(X)$, the mean $E[v(X)]$ and the correlation coefficient.

Any hints/directions will be appreciated, including a Bayesian approach.

Answer 1

No, you cannot say anything at all about $E[w(X)]$ from this.

Let $w_k(X)=w(X)+k$ for any constant $k$.

Then $v(X)$ and $w_k(X)$ have the same correlation $\rho$ as $v(X)$ and $w(X)$.

But this tells you nothing about $k$ or $E[w_k(X)]=E[w(X)]+k$, since $k$ can take any value.