# Random points on interval, expected lengths of pieces

Many years ago I came across the following task.

If we have the interval $$[0; 1]$$ and we throw $$N$$ uniformly distributed and mutually independent points on it, then we’ll get $$N+1$$ segments. What is the expected length of the longest segment? The 2nd longest? Etc.

For $$N=1$$, the solution is trivial: $$3/4$$ and $$1/4$$ (since the longest segment is uniformly distributed in [1/2; 1] and the shorter one is uniformly distributed in $$[0; 1/2]$$).

For $$N=2$$, the solution is not trivial, but possible. One just has to draw a quadrat 1 x 1. A point in it would mean that the longest segment has the x coordinate, and the 2nd longest segment has the y coordinate (and the shortest one is 1 – 1st – 2nd). One then has to carefully draw the possible area (this will be a triangle), and find its middle point.

But for $$N>2$$ I have no clue how to solve it.

I remember, the book I saw the task in, had a general solution for arbitrary $$N$$, but I don’t know anymore what book it was.

Note that the task is somewhat similar to Average Distance Between Random Points on a Line Segment, but just somewhat.