Confusion in definition of independent and identically distributed random variables

Question

From what I learnt, a random variable is a function which assigns real values to outcome space, and the probability distribution is a function that assigns probability to different values produced by random variable. Am I correct?
If yes, then what it means when we say two random variables are independent and identically distributed?

Answer 1

Take a random variable $X$ represented as \begin{align} X\,: &(0,1) \longmapsto \mathbb R\\ &~~~~u~~~\longmapsto X(u) \end{align} such that $$\mathbb P^{U\sim \mathcal U(0,1)}(X(U)\le x) = F_X(x)$$ Then if $U\sim\mathcal U(0,1)$, create $U_1$ made of the even bits of $U$ and $U_2$ made of the odd bits of $U$. Since $U_1$ and $U_2$ are independent, with identical Uniform distributions, $X(U_1)$ and $X(U_2)$ are iid from $F_X$.