Let $X$ be a random variable with probability distribution $f_X$
Then $f_X$ is a function,
$f_X : A \to B, c \mapsto d$
What exactly are $A,B,c,d$?
I am sorry if this question is basic, as I am from a computer science background and we do not usually specify these functions rigorously. In fact I am reading a paper right now and it says "let $p$,$q$, be probability distributions that generates data $X,Y$" - not very rigorous I tell you.