If $X$ and $Y$ are uncorrelated random variables, then under what condition is $E[X \mid Y] \approx E[X]?$ Suppose $X$ and $Y$ are real random variables that are uncorrelated. Now, uncorrelated does not imply independence, so $E[X \mid Y] \ne E[X]$.
However, can they be said to be approximately equal? If so, under what conditions does that approximation hold? (I realize that the equality is exact when $X,Y$ is a multivariate Gaussian, but I want to know more generally when the equality can be approximated.)
As a bonus, is it possible to do an expansion of $E[X \mid Y]$ that looks like "$E[X] + $higher order terms", so we can see explicitly when those higher order terms can be said to be negligible? Let's assume that $p(X \cap Y)$ is continuous and infinitely differentiable.


Now, uncorrelated does not imply independence, so $E[X \mid Y] \ne E[X]$.

I find the conclusion 'so $E[X \mid Y] \ne E[X]$' in this sentence a bit confusing.

*

*If variables are uncorrelated then it does not follow $E[X \mid Y] \ne E[X]$.

*In addition independence does neither mean $E[X \mid Y] \ne E[X]$. You can have independence while $E[X \mid Y] = E[X]$

If you have zero correlation then you can still have dependence, and also while $E[X \vert Y] = E[X]$
Example: Let $X \sim N(0,1)$ and $Y = N(0,\sigma^2 = X^2)$


If you have zero correlation then

*

*then this means that you have a slope of zero for a line that fits $E[Y|X]$ as function of $X$ or $E[X|Y]$ as function of $Y$.


*But $E[Y|X]$ can have all sorts of deviations from the straight line.
Example: let $X \sim N(0,1)$ and $Z = N(X^2,1)$



However, can they be said to be approximately equal?

In many situations, you have zero correlation, but still dependence due to heterogeneity as in the first example. There can be dependence but still $E[Y|X] = E[Y]$.
But it is difficult to give general conditions for this. The condition for $E[Y|X] = E[Y]$ is that $E[Y|X] = E[Y]$.
