On the Lengths of the Pieces of a Stick Broken at Random

摘要

The pieces of a randomly broken stick are considered. The length of the j the longest piece is determined and the exact distribution of the lengths is derived. It is noted that the analysis is particularly valid when the number of pieces is large. Also, moments of the lengths are obtained and some useful representations using independent exponential random variables are given. A necessary and sufficient condition for asymptotic normality is given and convergence of moments is obtained for linear combinations of length. The question of how many breaks are necessary to get all the pieces less than a certain length is also discussed. A limit distribution for the number of breaks is derived and convergence of moments is proved when the number tends to zero. The problem is also treated for the minimum number of arcs, having random length and placed at random on a circle of length one, needed in order to get complete coverage.