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

Let's draw two tables for the uniform r.v..

We define X and Y as below.

X = {0, 1, 2, 3}

Y = {0, 1, 2, 3}

Table 1

x f(x)

- ----

0 0

1 1

2 2

3 3

Table 2

x f(x)

- ----

0 3

1 2

2 1

3 0

Table 1 and Table 2 show that both f(x) are uniform r.v. because

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

but obviously f(x) of Table 2 is not an identity function because

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

but obviously f(x) of Table 2 is not an identity function because

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

However, we prefer to select an identity function for the uniform r.v., f(x).

So far, I answer the question myself because I have not found the answer on Internet yet otherwise I always be confused on uniform r.v. and identify function.

So far, I answer the question myself because I have not found the answer on Internet yet otherwise I always be confused on uniform r.v. and identify function.

-Count