Why don't we just stick to the mathematics involved. Here's a bit: Let f: R -> [0,oo) be a smooth probability density, and let X_1,...,X_n be independent random variables with this density. Define F(x) := Prob(X_j <= x) = Integral{-oo,x} f(u) du. Define M_n := max{X_1,...,X_n}. Then, Prob(M_n <= x) = F(x)^n. Call this G(x). So, the probability density g_n(x) of M_n is g_n(x) = (d/dx) F(x)^n = n F(x)^(n-1) f(x). (If the X_j are standard normal, we have f(x) = exp(-x^2/2)/sqrt(2 pi).) So, the expected value of M_n is E(M_n) = n Integral_{-oo,oo} u F(u)^(n-1) f(x) A little graph of this as a function of n appears on the page: < http://applet-magic.com/samplemax3.htm >, about 3/4 of the way down the page. --Dan On Jun 18, 2014, at 8:01 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.)