Will two dependent variables always have $\rho_{XY} \neq 0$?

Question

Let's say I have two random variables $X$ and $Y$ and I know: $$f_{XY}(x,y) \neq f_X(x)f_Y(y)$$

will these two random variables always have a correlation coefficient $\rho_{XY} \neq 0$?

I am aware of the following fact:

Independence implies $\rho_{XY} = 0$, but $\rho_{XY} = 0$ does not imply independence.

I am struggling to find any relationship regarding the opposite.

That is, does dependence imply $\rho_{XY} \neq 0$? I know I can check whether $$E[XY] = E[X]E[Y]$$ but I am wondering if this is needed.

Thanks!

Answer 1

The second part of the statement:
Independence implies $ρ_{XY}=0$, but $ρ_{XY}=0$ does not imply independence
is precisely equivalent to the fact that two dependent variables will NOT always have $ρ_{XY}\neq0$. It essentially says that there are variables that are not correlated but are not independent as well.