Uniform Random Variable v.s. Identify Function.

We know that a random variable r.v. is actually a function. Can we confirm that a uniform random variable must be an identity function? I don't think so. I also have not found that the identify function is a part of the definition of the uniform r.v. yet. Let me explain why.

Below is the r.v. definition:

X: U ---> V

Because r.v.  is also a function, I like write it as below only for easily thinking for me.

f: X ---> Y

The uniform r.v. is defined as below.

f: X ---> Y, Y = X
Pr [f(x) = y] = 1 / |X|

Below is the definition of identity function.

f: X ---> X
f(x) = x for all x is X

We define two random variables, f1 and f2, with same X and Y as below.

X = {0, 1, 2, 3}
Y = {0, 1, 2, 3}

f1: X ---> Y
f2: X ---> Y

Obviously, f1 and f2 are uniform r.v. because

Pr [f1(x) = y] = 1 / |X|
Pr [f2(x) = y] = 1 / |X|

but f2(x) is not an identity function because 

f2(x) <> x for all x is X.

However, we prefer to select the identity function, f1(x), as an uniform random variable.

So far, I answer the question myself because I have not found the answer on Internet yet. 

-Count