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I'm looking for a joint distribution $f(X, Y)$ over two real-valued variables $X$ and $Y$ which take on values between $0$ and $1$, such that $$ \forall x, y \in [0,1]^2 \quad f(X=x, Y=y) > 0 $$ and $$ E[X \mid Y=y] = y $$ and $$ E[Y \mid X=x] = x $$ Can anybody provide some simple distributions with these properties?

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    Example: $$P(X=Y)=1$$ – 2017-01-22
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    Thanks, maybe I should be more specific: I want the p.d.f. $f$ to be positive for all values of $x$ and $y$ inside the unit square. I've updated the question. – 2017-01-22
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    And what are the tries / thoughts / analogies that came to your mind? – 2017-01-22

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we have $$\mathbb{E}(X(X-Y))=\mathbb{E}(E[X(X-Y)\mid X]) = \mathbb{E}(X\times 0) = 0$$ Similarly, $$\mathbb{E}(Y(X-Y))=\mathbb{E}(E[Y(X-Y)\mid Y]) = \mathbb{E}(Y\times 0) = 0$$ Therefore $\mathbb{E}((X-Y)(X-Y)) = \mathbb{E}((X-Y)^2) = 0$, so $P(X=Y) = 1$, and it's impossible for $P(X = x, Y = y)$ to be positive for all $x$ and $y$.