Is this altered formula for correlation still bounded by $-1$ and $1$?

Question

Recall that

$$ ‐1 \le \text{corr}(X,Y) = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \le 1 $$

The proof for this bound uses the Cauchy Schwarz inequality, and I've been trying to wrap my head around why this bound holds intuitively.

Question: Does this bound still hold for the altered formula

$$ ‐1 \le \frac{\text{Cov}(X,Y)}{d_X d_Y} \le 1 $$

where

$$ d_X = E[|X - E[X]|] $$

and

$$ d_Y = E[|Y - E[X]|]? $$

I know that

$$ d_X d_Y \le \sigma_X \sigma_Y $$

so that, if it did hold, this would be a tighter bound than the original.

Answer 1

The bound doesn't hold for the altered formula. If $X$ is a standard normal variable, then $\operatorname{Cov}(X,X)=\operatorname{Var}(X)=1$ while $d_X= E|X| =\sqrt{\frac2\pi}$ (derivation here). Therefore $$\frac{\operatorname{Cov}(X,X)}{ d_X d_X}=\frac\pi2,$$ which is greater than $1$.

Answer 2

Let $X=-1$ with probability $2/3$, and $2$ with probability $1/3$, and let $Y=X$.

Compute. We have $d_X=d_Y=4/3$, and $\text{Cov}(X,Y)=2$. The ratio is $\frac{18}{16}$.