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Can a continuous random variable be jointly continuous with itself? Furthermore, is it always the case?

I don't see why not, but my issue is that $f_{XX}(x, x)$ looks weird. Does it accept two parameters, or only one?

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A couple $(X,X)$ is never jointly continuous.

This is because $P((X,X)\in D)=1$, where $D=\{(x,x)\mid x\in\mathbb R\}$, and the Lebesgue measure of $D$ is zero. In technical terms, one says that the measures $P_{(X,X)}$ and $\mathrm{Leb}_2$ are mutually singular.

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No, $X$ and $X$ are not jointly continuous: the pair $(X,X)$ always lies on the line $x=y$ in the $x,y$ plane.

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