Why does dependence of r.v.'s X and Y, not imply that X and Y are correlated.

Question

Given random variable X and Y.

Independence of X and Y, implies that X and Y are uncorrelated.

Why isn't the converse true:

dependence of X and Y, implies that X and Y are correlated?

Answer 1

Let $X= \begin{cases} -1 \\ \phantom{+}0 & \text{each with probability } 1/3. \\ +1 \end{cases}$

Let $Y=X^2.$

Obviously $X$ and $Y$ are not independent.

But they are uncorrelated.

As $X$ gets bigger, on average $Y$ does not get bigger or smaller.

Answer 2

Some examples have been given, but let us say one thing more.

We often think about correlation as measure of linear dependence. So:

• if $X,Y$ are independent, they are also not dependent by any linear relation.

• there can be no linear relation and still $X,Y$ can be dependent.

  $X,X^2$ where $X\sim \mathcal{N}(0,1)$ is a beautiful example.