This is perhaps too much of a general question, but perhaps someone can give me some insight.
Suppose that $X$ and $Y$ are random variables. Now, I'm interested in the joint distribution of $(X, X \mid Y)$.
I know that $$P(X, X \mid Y) = P([X \mid Y] \mid X) P(X) = P(X \mid [X \mid Y])P(X \mid Y) $$
Is it true then that $P(X, X \mid Y) = X P(X)$? What can one say about this joint distribution? If necessary, assume that $X$ and $Y$ are continuous and that $P(X \mid Y) = f(x\mid y)$ is the conditional density function.