So is this the solution? The probability that all numbers are less than x is equal to x^n. So then you take the derivative of that to get the probability that the maximum us is exactly x. n x^n-1. Then calculate the expected value as integral from 0 to 1 of x n x^n-1 = n/n+1.
Too lazy to think deeply about it, but it sounds right. You start with the cumulative distribution function and get the pdf from it, which looks like what you're doing.
Really simple solution. Problem is simple enough that this could be a standard HW problem in a probability course. Yet so many people (including myself) did not see it. We kept doing multiple integrals (n integrals for n points) and tried using induction on it.
This was actually a subproblem of the real problem. The real problem was: Given n points chosen randomly on a circle, construct the n-sided polygon. What is the probability that the center of the circle is inside the polygon? Since I worked on it, the problem has shown up on the Internet in various places (usually for the special case of n=3 - triangles, but I think I've seen the general one posted here and there). I don't recall if anyone came up with the same solution I did for the general case - I think one site had it.
it is (a standard HW problem). for bonus points, there's a connection between order statistics of the uniform distribution (such as max) and the beta distribution, of which the solution above is an example.
