Let $(\Omega, \mathcal{F} , \mathbb{P})$ be a probability space and $X:\Omega \to \mathbb{R}$ be a random variable.
When we simulate or pick a random sample of size $n$ from $X$, are we picking values $X (\omega_1), X (\omega_2), X (\omega_3), \dots, X (\omega_n)$ where each $\omega_i\in \Omega$, or are we picking values $X_1(\omega_1), X_2 (\omega_2), X_3(\omega_3), \dots, X_n (\omega_n)$ where the $X_i$s are iid random variables with the same distribution as $X$?
