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Suppose I have an uniform grid $A = [ a_1, a_2, a_3, \dots, a_n ]$ of $n$ points on the interval $[-c, +c]$.

As an example, consider $c=10$ and $n=10^4$ so that $$A = [-10, -9.9979998, -9.9959996, \dots, 9.9959996, 9.9979998, 10]$$

How should I randomly pick $m$ indexes $k_1, \dots k_m$ so that the resulting distribution of the sampled array $\tilde{A} = [a_{k_1}, a_{k_2}, \dots, a_{k_m} ]$ is approximately gaussian ?

I assume I'd need to compute the mean $\mu_A$ and std $\sigma_A$ of my original array $A$, and sample $68\%$ of my points in the interval $\mu_A \pm \sigma_A$, then $95\%$ in the interval $\mu_A \pm 2\sigma_A$... This is however not enough to approximate a gaussian.

Any ideas ?

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  • $\begingroup$ If indeed a uniform grid, $A$ is not a random sample. And generating a Gaussian over an integral is rigorously impossible since the Gaussian distribution is supported by $(-\infty,+\infty)$. $\endgroup$ Commented Mar 7, 2021 at 10:16
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    $\begingroup$ I'm not sure I understand your question: why can't you choose any Gaussian distribution with parameters $(\mu,\sigma)$ for which (say) $[\mu-3\sigma, \mu+3\sigma]\subset [-c,c]$, sample from it, and round the values to the nearest $a_i$? Because this doesn't require any information about the mean and sd of $A,$ nor anything about 68% and 95%, I wonder whether you have explained enough details of your problem to communicate it accurately. $\endgroup$ Commented Mar 7, 2021 at 16:43
  • $\begingroup$ @whuber this is good enough, if you post this as an answer I'll accept it ! $\endgroup$ Commented Mar 8, 2021 at 8:22

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