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lets look at my problem with this example:
$$ \int_0^1 f(x) dx=\mathbb{E}[f(U)]$$ where $U$ is uniformly distributed on $[0,1]$.

The Monte Carlo estimator would be $\displaystyle M:=\frac{1}{n} \sum_{i=1}^n f(U_i)$ for i.i.d. copies of $U.$

Then according to strong law of large numbers it would hold $\displaystyle \frac{1}{n} \sum_{i=1}^n f(U_i (\omega)) \rightarrow \mathbb{E}[f(U)]$ for "almost all omega".

But when simulating it on computer what one does is to sample n different $\omega$ as in $\displaystyle \frac{1}{n} \sum_{i=1}^n f(U (\omega_i))$.
I am not quite sure why this is the same.

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  • $\begingroup$ You mean "according to the strong law of large numbers". $\endgroup$ – kimchi lover Sep 24 '18 at 18:02
  • $\begingroup$ Yeah sure I mixed that up, gonna fix it. The question still standing tho $\endgroup$ – StefanWK Sep 24 '18 at 21:38

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