Here are a couple of simple statistical questions that I don't know the answers to, that have bearing on what we would expect to see in the outliers of a distribution. Suppose we take n independent samples from a normal distribution (let's fix the distrubution as having mean 0 and variance 1). As n increases, what happens to the expected value of (largest sample - 2nd largest sample)? How sensitive is this answer to the normality of the distribution? What do we need to know about the distribution to conclude that as n increases, this difference will behave the way it does for a normal distribution? Andy On Wed, Jun 18, 2014 at 11:01 AM, James Propp <jamespropp@gmail.com> wrote:
The question Dan Asimov raised (which I paraphrase as "How are outliers distributed?") strikes me as one that's well-suited to math-fun, in at least two ways: it's an interesting mathematical question, and it's an interesting question about mathematicians!
Then again, raising questions about the distribution of achievement at the high end of the talent-spectrum in STEM fields is the kind of thing that college presidents get fired for. So if anyone chooses to follow up on the second issue I mention in this message, I hope we can keep the discussion thoughtful and civil. (Rich, feel free to pre-emptively shut down this topic if based on your experience you feel it's unlikely to lead to anything other than elevated cortisol levels.)
Jim Propp
On Wed, Jun 18, 2014 at 10:12 AM, Charles Greathouse < charles.greathouse@case.edu> wrote:
I was talking about the secretary problem with valuations 1, 2, ..., N, not the problem of hiring an employee in real life. (I don't know why we'd discuss the latter on math-fun.)
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Wed, Jun 18, 2014 at 9:40 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I don't think this is clear at all. Take an individual sport where little chance is involved, like tennis or pocket billiards: often there is one person dominating the number one spot for years on end.
--Dan
. . . the difference between best and second-best (say) is very small when N is large.
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