Let $X$ be a random variable, and $f$ a measurable function. Is there any particular relationship between the expression of $f$ and $corr(f(X),X)$?

###BACKGROUND###

The background of asking the value of $corr(f(X),X)$ is as following.

From the book on elementary statistics, I learned the conditional expection $E[Y|X]$ is the best approximation to the random variable $Y$, under the criterion of minimizing the $E[Y-L(X)]^2$.

Denote by $M(X)$ the $E[Y|X]$, and assume $(X,Y)$ subject to the multivariable normal distribution with parameters(mean, variance and correlation) $u_1,u_2,\sigma_X^2,\sigma_Y^2,\rho$.

In this case, ` $M(X)=u_2+\frac{\rho \sigma_2}{\sigma_1}(X - u_1)$. And computation shows $$ corr(M(X),X) =\frac{ \sigma_X}{ \sigma_M} M'(0) $$. So this seems a nice relation.

QUESTION

So I just want to know, is it possible to extend the above result to general $(X,f(X))$?

Or, if we restrict $f$ to a special class of functions, is there any nice result on $corr(X,f(X)$? I tried write $f(X)$ as a power series, but came up with nothing interesting.

If there are some results on it, where can I find the references?

Thank you.

Let $X$ be a random variable, and $f$ a measurable function. Is there any particular relationship between the expression of $f$ and $corr(f(X),X)$?

BACKGROUND

The background of asking the value of $corr(f(X),X)$ is as following.

From the book on elementary statistics, I learned the conditional expection $E[Y|X]$ is the best approximation to the random variable $Y$, under the criterion of minimizing the $E[Y-L(X)]^2$.

Denote by $M(X)$ the $E[Y|X]$, and assume $(X,Y)$ subject to the multivariable normal distribution with parameters(mean, variance and correlation) $u_1,u_2,\sigma_X^2,\sigma_Y^2,\rho$.

In this case, ` $M(X)=u_2+\frac{\rho \sigma_2}{\sigma_1}(X - u_1)$. And computation shows $$ corr(M(X),X) =\frac{ \sigma_X}{ \sigma_M} M'(0) $$. So this seems a nice relation.

QUESTION

So I just want to know, is it possible to extend the above result to general $(X,f(X))$?

Or, if we restrict $f$ to a special class of functions, is there any nice result on $corr(X,f(X)$? I tried write $f(X)$ as a power series, but came up with nothing interesting.

If there are some results on it, where can I find the references?

Thank you.