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So I have some code simulating the behavior of a complex system. The system works perfectly (with a "fidelity" of $f=1$) for some parameter $\beta=0$. For other values of $\beta$ the fidelity will be some number between 0 and 1. I know that $f(\beta)$ peaks for $\beta=0$ (not sure how sharply exactly) and then drops off to some kind of noisy floor for larger absolute values of $\beta$. I don't have any analytic insight into $f(\beta)$ as a function.

Now, if I draw $N$ values $\beta$ from a Gaussian distribution with mean 0 and a fixed standard deviation $\sigma_\beta$, and then calculate the mean fidelity as $\bar f(\sigma_\beta) = \frac{1}{N}\sum_{n=1}^N f(\beta_n)$, is there a way for me to estimate the standard error on $\bar f$? Or, alternatively, is there a good way for me to figure out how many samples $N$ I need to take to get a value of $\bar f$ that I feel confident about within 1%?

This might be a fairly basic question, but it's been a long time since my undergraduate days ;-). In the absence of anything quantitative, I'd probably just try pulling 100 samples and check how stable the resulting $\bar f$ is compared to if I just used 50 samples (i.e., throw out every other sample).

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