Condition for independence of two scalar real valued random variables

Question

I'm trying to show that given two real-valued scalar random variables $X,Y$ if $$\mathbb{P}\left(X\leq x,\, Y\leq y\right)\cdot\mathbb{P}\left(X\geq x,Y\geq y\right)=\mathbb{P}\left(X\geq x,Y\leq y\right)\cdot\mathbb{P}\left(X\leq x,Y\geq y\right)$$ holds for every $\left(x,y\right)\in\mathbb{R}^{2}$ then $X,Y$ are independent. I tried going at it from the definitions but got a bit stuck, I'd appreciate some help.

Answer 1

$P_{X,Y}\left(x,y\right)\left[1-P_{X}\left(x\right)-P_{Y}\left(y\right)+P_{X,Y}\left(x,y\right)\right]=\left[P_{Y}\left(y\right)-P_{X,Y}\left(x,y\right)\right]\left[P_{X}\left(x\right)-P_{X,Y}\left(x,y\right)\right]$

(left-side=right-side)

leads to:

$P_{X,Y}\left(x,y\right)=P_{X}\left(x\right)P_{Y}\left(y\right)$ (independence).

Here $P_{X}\left(x\right)=P\left\{ X\leq x\right\} $, $P_{Y}\left(y\right)=P\left\{ Y\leq y\right\} $ and $P_{X,Y}\left(x,y\right)=P\left\{ X\leq x,Y\leq y\right\} $.

To make things precise every $\ge$ in your question should be replaced by $>$.