Does independence of two random variables imply uncorrelatedness?

Question

There are many materials about the reverse question: "Does uncorrelatedness tell us something about independence?" But how to answer the question I've posed and why? Is there some simple counterexample?

Answer 1

I will try to give an intuitive answer.

To say that two random variables $X$ and $Y$ are uncorrelated means that $\mathbb{E}[XY] = \mathbb{E}[X] \mathbb{E}[Y]$. On the other hand, saying $X$ and $Y$ are independent means that their joint distribution is the product of their marginal distributions, $p(X,Y) = p(X) \cdot p(Y)$; or, equivalently, that either joint distribution is the same as its marginal distribution $p(Y|X) = p(Y)$.

Now, suppose that $X$ and $Y$ are correlated. Then there must be values $x_{1}$ and $x_{2}$ for which $\mathbb{E}(Y|X=x_1) \neq \mathbb{E}(Y|X=x_2)$.*

Since the expectations are unequal, it is clear that $p(Y|X=x_1) \neq p(Y|X=x_2)$. But if $X$ and $Y$ were independent, we would have $p(Y|X=x_1) = p(Y) = p(Y|X=x_2)$. Thus correlation implies dependence and, by the contrapositive, independence implies uncorrelation.

* This is easiest to see by assuming that $\mathbb{E}[Y]$ is constant and working out $\mathbb{E}[XY]$ by integration