Terminology: Independent Copy of Random Variables Suppose $X_1,\ldots, X_n$ are (independent) RVs. What does it mean to say that $X_1',\ldots, X_n'$ is an independent copy of $X_1,\ldots, X_n$?
Does it mean that each $X_i'$ is independent of $X_i$ or does it mean that the joint distribution of $(X_1,\ldots, X_n)$ is the same as the joint distribution of $(X_1',\ldots, X_n')$? Or does it mean something else entirely? 
I find the term a bit confusing since I am not sure how you can be both independent and a copy (since being a copy would imply being dependent).
 A: Consider the case when $n=1$. When we say that $X'$ is an independent copy of $X$ what we mean is that the distribution of $X'$ is the same as the distribution of $X$ AND that $X$ and $X'$ are independent. (you can use any other symbol for $X'$). This expression is usually used when you want to repeat the same exact statistical experiment number of times independently. So, people tend to use the same symbol with a prime or tilde.
For example one use for Monte Carlo simulations,is to approximate the expectation of a function $\varphi(X_1, X_1, \dots, X_n)$ in which the joint random variable
\begin{equation}
(X_1, X_1, \dots, X_n) \sim P_X
\end{equation}
To do this, we need the realizations of number of independent copies of $(X_1, X_1, \dots, X_n)$. In your question you only have 2 copies. In Monte Carlo methods, we should use as many copies as possible. The estimator of the expectation is then given by
\begin{equation}
{ \hat{\mathbb{E}}}[\varphi(X_1, X_2, \dots, X_n)]: = \frac{1}{M} \sum_{m=1}^M \varphi(X^{(m)}_1, X^{(m)}_2, \dots, X^{(m)}_n)
\end{equation}
in which 
\begin{equation}
X^{(m)}_1, X^{(m)}_2, \dots, X^{(m)}_n \quad \text{for all } m \in \{1, \dots ,M\}\backslash i \quad \text{for some } i \in \{1, \dots ,M\}
\end{equation}
are independent copies of a random variable
\begin{equation}
X^{(i)}_1, X^{(i)}_2, \dots, X^{(i)}_n \sim P_X.
\end{equation}
Observe that 
\begin{equation}
{ \hat{\mathbb{E}}}[\varphi(X_1, X_2, \dots, X_n)]
\end{equation}
is a random variable that depends on the $M$ copies. Once you realize all the copies, you get an estimate of the expectation.
