What information does one need to determine if two random variables are independent? Suppose I have two random variables with unknown distributions. I understand the definition of independence is 
$$P(X=x,Y=y) = P(X=x)P(Y=y)$$
$$f_{XY}(x,y) = f_X(x)f_Y(y)$$
My Question 
Besides giving the distributions themselves, is there another way to deduce that two random variables are independent? Or is the only way to have the distribution a priori and then prove two random variables are independent explicitly? 
 A: As always, to prove that something has a property you have to go back to the definition of that property either directly, or indirectly by applying theorems that refer to that property. So probably your question should be taken to ask what theorems there are to conclude that two random variables are independent.
Math1000 gave such a theorem in a comment. Another one is that functions of independent variables are independent, e.g. if $X$ and $Y$ are independent, then so are $X^2$ and $Y^3$. Another one is that $X$ and $Y$ are independent iff the characteristic function of the random vector $(X,Y)$ satisfies $\varphi_{(X,Y)}(t,s)=\varphi_X(t)\varphi_Y(s)$.
An extra-mathematical way of arriving at a belief that two random variables are independent is to conclude that there's no physical mechanism for the quantities they represent to be related – for instance if you roll two dice without coupling them.
A: On Top to what mentioned above. $X$ and $Y$ are independent if knowing the value of one doesn't effect the value of the other In terms of conditional and marginal distribution. saying that the $P(Y|X=x) = P(Y)$
To generalize your definition we can say. $P(X \in A, Y \in B) = P(X \in A)P(Y \in B)$ for all subsets $A$ and $B$ of real numbers
