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I am trying to find an example of two random variables with joint CDF: $F(x, y) = \mathcal{P} ( X\leq x, Y \leq y) = I_{(x+y\geq 1)}.$

I can say that there is one such vector because given CDF is actually a CDF i.e. it satisfies all required conditions. And there is a theorem that states that in that case a random vector with given CDF exists. Is my statement correct?

It is obvious that $X$ and $Y$ are dependent and this CDF seems so simple yet I can't think of any example.

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    I have corrected my question, thank you for your comments.2012-10-13

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The "joint CDF" you propose is not possible. It would give us

$P(X\le 2,Y\le 0) = P(X\le 0,Y\le 2) = 1$

so therefore $(X,Y)$ would almost certainly satisfy each of $Y\le 0$ and $X\le 0$. But the same joint CDF claims that

$P(X\le 0, Y\le 0)=0$

which is a contradiction.