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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 $X$?

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  • $\begingroup$ Related: Mathematical description of a random sample $\endgroup$ – Artem Mavrin Apr 16 '16 at 23:56
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    $\begingroup$ The right option is a third one: we are picking values $X_1(\omega)$, $X_2 (\omega_)$, $X_3(\omega)$, $\dots$, $X_n (\omega)$, where the $X_i$s are i.i.d. random variables with the same distribution as $X$. $\endgroup$ – Did Apr 17 '16 at 7:49
  • $\begingroup$ Thanks guys. @Did, which space would that 'omega' belong to? Is it the product space ? $\endgroup$ – SuperM Apr 17 '16 at 15:41
  • $\begingroup$ No, to the set Omega each random variable is defined upon. $\endgroup$ – Did Apr 17 '16 at 21:11

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