Joint PDF of X and a Function Y = g(X)

#1
Thakur here,

My question is regarding the joint pdf of the input and output of a system. Lets say I have a system represented by a deterministic function Y = g(X).

Lets say X is uniformly distributed.

I know I can find pdf of Y "p(y)" using standard textbook techniques. However, is it possible to find the joint (or conditional) distribution of X and Y given the knowledge of p(X) and g(X)..?

Or would it be fair to say that p(y) represents the conditional distribution of Y given X.

Pointers to worked examples would be very helpful.

Thanks in advance.
 
#2
Ok...after doing some thinking, I think I have realized that given x; I know Y with Probability of 1 and there is no randomness. Therefore in this case, p(Y|X) is basically meaningless. It would be relevant is the system was random and had some noise which added uncertainty about given the value of X.