# Is sampling according to several i.i.d. random variables or to just one of them equivalent

Suppose we have n+1 random variables i.i.d. distributed $X_0,X_1,...X_{n}$. Is it the same operation if I generate n samples according to $X_0$ $\{S^0_1,S^0_2..,S^0_n\}$, or if I take from each of the random variables $X_1,...,X_{n}$ a sample, thus $\{S^1_1,S^2_1..,S^n_1\}$. The reason for asking is because I am implementing a version of Monte Carlo simulation. I would like to know if there is such equivalence, So I could implement it in two manners. This question may look obvious, but these two approaches are (maybe) not similar for sample generation. Thank you.

• Both methods will give the same result. – Dilip Sarwate Feb 28 '15 at 14:23
• BY definition, a random variable $X_0$ is a measurable transform from the sampling space to $\mathbb{R}$. Strictly speaking, if you follow this definition, the $n$ transforms $X_0(\omega),\ldots,X_0(\omega)$ are then identical. Obviously, from a Monte Carlo perspective, this does not matter as $X_0$ is a transform of a uniform and hence transforming $n+1$ uniforms by $X_0$ or by $X_0,\ldots,X_n$ ends up with the same distribution. – Xi'an Feb 28 '15 at 14:31