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

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.

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

-Count