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To keep things simple let's assume you have a perfect random number generator (i.e. I will discuss only the statistics not the numerics of the problem). I will also focus on the practical matter and gloss over some mathematical details. From a practical perspective "convergence" means that you will never get an exact answer from Monte-Carlo but ...


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the output of an MC simulation depends on the random numbers used and if the distribution used is not too weird, after 10,000 runs you will get an answer that is distributed $$ \mu + \frac{\sigma}{\sqrt{n}} Z, $$ with $Z$ a standard normal. Here $n=10,000.$ With $\mu$ the quantity you want and $\sigma$ the standard deviation. So you won't get precisely the ...



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