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.


1 Answer 1




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 $>$.


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