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Suppose $X \sim U(0,1)$ and $Y$ and $Z$ are random variables that depend on $X$. I've solved a problem where $Y$ and $Z$ are discrete (binary) and so finding the joint pmf just amounts to calculating probabilities for a finite set of possibilities. However it got me thinking, how would i calculate the joint pdf if $Y$ and $Z$ are continuous? I.e imagine if we had a scenario where $Y \sim U(0.3,x)$ and $Z \sim U(0.7,x)$ (sorry for abuse of notation not looking at the order of $x$ and $0.3$ and $0.7$ respectively.) How would we calculate the pdf of $f_{Y,Z}(y,z)$. Would we first calculate the joint CDF or something?

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The model is not completely specified: if $Y$ and $Z$ are independent given $X$ then $$(X,Y,Z)\sim f(x) g(y|x) h(z|x)$$ else $$(X,Y,Z)\sim f(x) g(y|x) h(z|x,y)$$ where $f$, $g$, $h$ are the appropriate conditional densities.

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