What is the maximum of $n$ points with CDF $F$ and PDF $f$?

Question

I read somewhere that the minimum of $n$ points with CDF $F$ and PDF $f$ is

$g(y) = n(1-F(y))^{(n-1)}f(y)$

What would the corresponding maximum value of the points be? Also, how do we derive the minimum and maximum values?

@Chinny84, I understand that's how we can get the minimum value. So, how about the maximum value? — saikat, Apr 21 '15 at 20:21
Oops I made a mistake (not for the first time recently it seems) I will take a further look (now I read your question properly) — Chinny84, Apr 21 '15 at 20:23

score 0 · Answer 1 · answered Apr 21 '15 at 22:12

Ok, so I figured there's a general formula for the $k^{th}$ order statistic.

$g_k(y_k) = \frac{n!}{(k-1)!(n-k)!}[F(y_k)]^{k-1}[1-F(y_k)]^{n-k}f(y_k)$

So, the maximum value can be obtained by setting $k=n$ and the minimum can be obtained by setting $k=1$.

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