Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-30T20:09:58.974Z Has data issue: false hasContentIssue false
  • English
  • Français

On the lengths of the pieces of a stick broken at random

Published online by Cambridge University Press:  14 July 2016

Lars Holst
Lars Holst*
Affiliation:
Uppsala University
*
∗Postal address: Department of Mathematics, Uppsala University, Thunbergsvägen 3, S-752 38 Uppsala, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider the pieces of a randomly broken stick. How long is the j th longest piece? How many breaks are necessary for getting all pieces less than a given length? These and related questions are studied in particular when the number of pieces is large. Using simple properties of the exponential distribution new proofs are given of old results and new results are obtained.

Keywords

SPACINGSUNIFORM DISTRIBUTIONEXPONENTIAL DISTRIBUTIONCOVERING A CIRCLEORDER STATISTICSLIMIT THEOREMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 3 , September 1980 , pp. 623 - 634
Copyright
Copyright © Applied Probability Trust 1980 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Darling, D. A. (1953) On a class of problems relating to the random division of an interval. Ann. Math. Statist. 24, 239–253.Google Scholar
Edens, E. (1975) Random covering of a circle. Indag. Math. 37, 373–384.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and Its Applications , Vol. 2, Wiley, New York.Google Scholar
Fisher, R. A. (1929) Tests of significance in harmonic analysis. Proc. R. Soc. London A 125, 54–59.Google Scholar
Fisher, R. A. (1940) On the similarity of the distribution found for tests of significance in harmonic analysis and in Stevens' problem in geometrical probability. Ann. Eugenics, London 10, 14–17.Google Scholar
Flatto, L. (1973) A limit theorem for random coverings of a circle. Israel J. Math. 15, 167–184.Google Scholar
Flatto, L. and Konheim, A. (1962) The random division of an interval and the random covering of a circle. SIAM Rev. 4, 211–222.CrossRefGoogle Scholar
Hecker, H. (1976) A characterization of the asymptotic normality of linear combinations of order statistics from the uniform distribution. Ann. Statist. 4, 1244–1246.CrossRefGoogle Scholar
Holst, L. (1980) On multiple covering of a circle with random arcs. J. Appl. Prob. 17, 284–290.CrossRefGoogle Scholar
Kaplan, N. (1977) Two applications of a Poisson approximation for dependent events. Ann. Prob. 5, 787–794.Google Scholar
Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin, London.Google Scholar
Lecam, L. (1958) Un théorème sur la division d'un intervalle par des points pris au hasard. Publ. Inst. Statist. Univ. Paris 7, 7–16.Google Scholar
Levy, P. (1939) Sur la division d'un segment par des points choisis au hasard. C. R. Acad. Sci. Paris 208, 147–149.Google Scholar
Pyke, R. (1965) Spacings (with discussion). J. R. Statist. Soc. B 27, 395–449.Google Scholar
Pyke, R. (1972) Spacings revisited. Proc. 6th Berkeley Symp. Math. Statist. Prob. 1, 417–427.Google Scholar
Rao, J. S. (1976) Some tests based on arc-lengths for the circle. Sankhya B 33, 1–10.Google Scholar
Rao, J. S. and Sethuraman, J. (1975) Weak convergence of empirical distribution functions of random variables subject to perturbations and scale factors. Ann. Statist. 3, 299–313.Google Scholar
Shepp, L. A. (1972) Covering the circle with random arcs. Israel J. Math. 11, 328–345.Google Scholar
Steutel, F. W. (1967) Random division of an interval. Statist. Neerlandica 21, 231–244.Google Scholar
Stevens, W. L. (1939) Solution to a geometrical problem in probability. Ann. Eugenics, London 9, 315–320.Google Scholar
Whitworth, W. A. (1897) Exercises on Choice and Chance. Deighton Bell and Co., Cambridge. (Republished by Hafner, New York (1959).).Google Scholar