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Let $x$ and $y$ be 2 realizations of the continuous RVs $X$ and $Y$, such that $X,Y\sim\text{Uniform}(0,1)$. Given $x$ (or $y$), if I know that $y$ (or $x$) is bounded by $y>1-x$, does it mean that $Y>1-X$? If so, that would demonstrate that $F_{X,Y}(x,y)=0$ for pairs $(x,y)$ such that $y<1-x$, wouldn't it?

Cromack
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  • What is the quantifier in the "bounded by" statement: does it refer to *this particular realization* $(x,y)$ or to *all possible realizations?* – whuber Oct 23 '19 at 14:31
  • @whuber Imagine I have a realization of the form $(x,y)$. If I could just know the value of $x$, I would know that $y>1-x$. And the same happens for every possible pair. – Cromack Oct 23 '19 at 15:01
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    That is *precisely* what the relation $Y \gt 1-X$ means. – whuber Oct 23 '19 at 15:30

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For background: Is that meaningful to compare two random variables?, in particular, "If X and Y are random variables with values on the real line or any other space that can be ordered then {X>Y} has meaning as a measurable event."

If $y>1-x$ for all x and y, then $Y>1-X$ is an event with the property $P(Y>1-X)=1$.

If $y<1-x$ then $F_{X,Y}(x,y)=P(X\leq x,Y\leq y)=0$ as you thought.

user133631
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  • Thank you. As a side note (and unfortunately), we cannot say much about $F_{X,Y}(x,y)$ if $y>1-x$, can we? – Cromack Oct 24 '19 at 07:26