# Interesting variant on discrete probabilistic problem

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?

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.