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There is a line segment of the length of $1$. $N-1$ points are randomly chosen in it, so it is divided by $N$ parts. The question is to calculate expected values of all these parts, from the shortest one to the longest one. I know that the correct answer for the length of the shortest part is about $1/N^2$, but I have no idea how to find out the whole distribution of lengths.

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Since the distribution is uniform this can be considered the search for the centroid of a figure in Euclidean space.

A priori the figure to consider for $n-1$ independent uniform variables would be the $(n-1)$-dimensional hypercube, but we are interested in the differences (lengths of the segments) so we study the $(n-1)$-simplex defined in $n$-dimensional space by the constraints that all $n$ coordinates be nonnegative, and that $\sum_{i=1}^nx_i=1.$ Thus the coordinates do not represent the original variables, but their differences (with appropriate interpretations at the boundaries).

We should divide this into $n!$ subsets according to the order in which the lengths of the line segments appear; however, by symmetry the same effect can be achieved by limiting our attention to the one subset where we already have

$$x_1\geq x_2\geq\ldots\geq x_n.$$

That subset is also an $(n-1)$-simplex and its $n$ vertices are

$$\matrix{ (1,0,0,\ldots,0,0,0)\\ (\frac12,\frac12,0,\ldots,0,0,0)\\ (\frac13,\frac13,\frac13,\ldots,0,0,0)\\ \vdots\\ (\frac1{n-1},\frac1{n-1},\frac1{n-1},\ldots,\frac1{n-1},\frac1{n-1},0)\\ (\frac1n,\frac1n,\frac1n,\ldots,\frac1n,\frac1n,\frac1n) }$$

The centroid is just the average of those $n$ vertices, so by taking the sum of each column in the above list and dividing by the number of vertices we get that the $x_k$-coordinate of the centroid is

$$\frac1n\sum_{i=k}^n\frac1i.$$

This is the expected value of the length of the $k$-th line segment in decreasing order.

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