Correlation vs Conditional Expectation I have two random variables, $X$, and $Y$. What are the differences or similarities between Corr$(X, Y)$ and $\mathbb{E}(X|Y=y)$. Can I deduce one from the other?
 A: In general, it's not possible to tell what is the exact relationship between the correlation and conditional expectation unless $X$ and $Y$ are assumed to be jointly normal. Thus, here I'll mostly focus on

What are the differences or similarities ... ?

Definitions first. When $X$ and $Y$ are both random variables, $\text{cov}(X,Y) = E(XY) - E(X)E(Y)$, which, can also be written as (see Henry's comment)
$$
E_Y(Y E_X(X∣Y))−E_X(X)E_Y(Y).\tag{*}
$$
This is the only thing we can tell in general, the covariance does depend on the conditional expectation. To tell "how", we need to know further about the joint distribution of $X,Y$.
Now the covariance is a real number that measures the degree of linear association between $X$ and $Y$; $\text{cor}(X,Y)$ is just a scaled version of $\text{cov}(X,Y)$, which does not have a unit of measurement. By (*) we also see that $\text{cov}(X,Y)$ depends on $E(X|Y=y)$, the conditional expectation of $X$ given $Y=y$.
On the other hand, for a fixed value $y$ of $Y$, $E(X|Y=y)$ is a measure of the location of $X$. Thus $E(X|Y=y)$ is a measure of the location for $X$ conditionally on $Y$.
Thus the correlation

tells us how $X,Y$ vary together and it does not depend on quantities other than the distribution of $Y,X$.

The conditional expectation $E(X|Y=y)$

tells us how the population average of $X$ moves along its axis as $Y$ moves in its own space and it does depend on extra quantities, here $y$.

Example 1. Suppose $Y$ and $X$ are independent random variables and $X$ has an expected value different from zero, i.e. $E(X)\neq0$. Then,
$$\text{cov}(X,Y) = E(XY)-E(X)E(Y) = E(X)E(Y)-E(X)E(Y)=0.$$
On the other hand,
$$E(X|Y=y) = E(X) \neq 0.$$
The conclusion is that, $\text{cov}(X,Y)$ coincides with $E(X|Y)$ only if $Y$ and $X$ and independent and $E(X)=0$.
For further intuition, let's consider another example.
Example 2.  Suppose $X$ measures life expectancy and $Y$ measures the age of an individual. I expect $X,Y$ to be correlated negatively, since the larger the age of an individual the smaller its life expectation, certeris paribus. On the other hand, for a given age say 50 years, e.g. $Y=50$, I expect my sub-population to take variable life expectancy values; Some people have higher life expectancy than others for many reasons. The conditional expectation tells us what is the most reasonable life expectancy for this sub-population.
Note that when $X$ and $Y$ are random vectors $\text{cov}(X,Y)$ and $E(X|Y=y)$ have totally different dimensions and are not comparable. Indeed, if $X$ is a random vector of size $m\times 1$ and $Y$ is a random vector of size $n\times 1$, then $\text{cov}(X,Y)$ is an $m\times n$ matrix whereas $E(X|Y=y)$ is an $m\times 1$ vector.
