Let S be a set of N numbers. Permute them into random order. Scan thru the elements in that order. Each time you encounter a new record max, ring a bell. So for example, if S={7*,8*,5,3,4,9*} you would ring the bell 3 times at the *s.
EASY CLAIM: expected number of times the bell rings, equals HarmonicNumber(N) = 1+1/2+1/3+...+1/N.
HARDER QUESTIONS, which I have not looked at: What about the variance? What about tail bounds? Can you write a formula for the whole probability distribution for the number of bell-rings?
--ok, thanks to Victor Miller's hint, we have expected # bell rings = ha(N) = 1 + 1/2 + 1/3 + ... + 1/N. probability of exactly R rings = |s(N,R)| / N! where s(N,R) = Stirling Numbers of the first kind. variance of #bell rings = SUM(R=0..N)OF [R-ha(N)]^2 * |s(N,R)|/N! Of course, when N is large, ha(N) approaches ln(N)+gamma where gamma= 0.5772156649 is the Euler-Mascheroni constant. The variance appears numerically to approach ha(N)-1.645 which if so would mean that the probability distribution is sharply peaked around its expectation, e.g. when N--> infinity the probability-->100% that #bellrings lies within the interval [0.999, 1.001]*ha(N). That's remarkably predictable.