[HoneyBadger]

I don't know if this has been asked, but I'll do it anyway:

Say you are given random numbers consecutively from the set of reals, and you are given N (fixed) of these numbers in total, one by one. At any moment you can stop and "win" the current number as score. The objective is to get the highest score.

Is there an optimal strategy?
How does it depend on N?
Does it matter when there is an upper or lower bound to the number range?

(I guess you can see where this is heading... is this a "hard" problem? I suppose it must be documented in economics literature)