Solved – Does non-zero correlation imply dependence

Question

We know of the fact that zero correlation does not imply independence. I am interested in whether a non-zero correlation implies dependence – i.e. if $\text{Corr}(X,Y)\ne0$ for some random variables $X$ and $Y$, can we say in general that $f_{X,Y}(x,y) \ne f_X(x) f_Y(y)$?

Answer 1

Yes, because

$$\text{Corr}(X,Y)\ne0 \Rightarrow \text{Cov}(X,Y)\ne0$$

$$\Rightarrow E(XY) - E(X)E(Y) \ne 0 $$

$$\Rightarrow \int \int xyf_{X,Y}(x,y)dxdy -\int xf_X(x) dx\int yf_Y(y)dy \ne 0$$

$$\Rightarrow \int \int xyf_{X,Y}(x,y)dxdy -\int \int xyf_X(x) f_Y(y)dxdy \ne 0$$

$$\Rightarrow \int \int xy \big[f_{X,Y}(x,y) -f_X(x) f_Y(y)\big]dxdy \ne 0$$

which would be impossible if $f_{X,Y}(x,y) -f_X(x) f_Y(y) =0,\;\; \forall \{x,y\}$. So

$$\text{Corr}(X,Y)\ne0 \Rightarrow \exists \{x,y\}:f_{X,Y}(x,y) \ne f_X(x) f_Y(y)$$

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