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I was wondering if there was an analytic description of the distribution of the largest $m$ of $n$ samples of a Gaussian distribution, where $n \geq m$.

(As an example, I generated 100 samples from $\mathcal{N}(\mu = 0, \sigma = 1)$. The average of the top 100 samples came to $0.0404$, the average of the top 50 samples came to $0.832$, the average of the top 10 samples came to $1.842$, and the top 1 sample was 2.88.)

Intuitively, as $n$ increases and $m$ stays constant, or $m$ decreases and $n$ stays constant, the expected value increases. It's like taking the top $m$ applicants.

Is there a name or analytic description of such a distribution?

(I haven't been able to figure out how to do this, and I spent a lot of time searching. This seems like a simple / common problem and I'm surprised I haven't found anything on it.)

EDIT: After the comments below, I as able to find a good answer here:

Approximate order statistics for normal random variables

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    $\begingroup$ What do you mean top 𝑚 of 𝑛 samples? Are you saying you sorted the n elements, and pick the highest m elements? $\endgroup$
    – Bill Chen
    Nov 7, 2019 at 18:49
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    $\begingroup$ The key work is order statistics and you can find the answer on the Wikipedia page. $\endgroup$
    – Xi'an
    Nov 7, 2019 at 18:52
  • $\begingroup$ @ BillChen Exactly right! I mean the largest $m$ of $n$ samples. Sort the $n$ elements then pick the highest $m$. @ Xi'an This looks related to what I'm looking for! From there I was able to find "sampling distribution". I'll edit my question / mark as a duplicate (was able to find a relevant stats.SE post). Thank you! EDIT: I'm dumb, it seems a "flag" is more of a moderation tool, and I probably should have edited my post instead of flagging it. $\endgroup$
    – lynn
    Nov 7, 2019 at 18:59

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