A friend of mine recently posed an interesting question to me. We can easily see that if we choose a random number between 0 and 1, the expected value should be one-half. Now, in this case, the only number is the same as the largest number, so that the expected value of the largest number is also one-half.
So what occurs if we choose two random numbers? Can we determine the expected value of the largest number in this case? What if there are N numbers? I work through a pretty direct interpretation for the N=2 case, then find a general approach for arbitrary N.
Given the result of N/(N+1), I noticed that this is equal to 1 minus the integral from 0 to 1 of x^N. This would be a really quick solution, but I can't seem to find an interpretation of the situation which lends itself nicely to that expression. Perhaps it's a coincidence, but who knows?
