I have a question regarding the Monte Carlo integration method. This is probably a "noob" question, but I have searched the internet and haven't been able to find an answer…
Let's say I want to estimate an integral of some function over an area $D$, that I'll denote as $I$. I can choose $N$ points to randomly sample from $D$ in order to obtain the integral, the error should "decay" as $err\sim \frac{\sigma}{\sqrt{N}}$, so that $I\sim \overline{I}\pm\frac{\sigma}{\sqrt{N}}$. But what if I just ran the simulation $N$ times and would get different values like: $I_1,I_2,….I_N$ those should be distributed normally as well, right? So I can estimate the error from those values again as $I\sim\overline{I} \pm \frac{\sigma}{\sqrt{N}}$, would this give the same results or is my thinking wrong?
Best Answer
As rightly stressed in the comments of J Bowman, there is a fair amount of confusion in this question. Considering