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Random points associated with rectangles

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Abstract

The study of the distribution and moments of the distance between random points within a rectangle or in two coplanar rectangles is required in a wide variety of fields. Formulae for the distributions and arbitrary moments of the distance between two random points associated with one or two rectangles in various situations are given here explicitly. These explicit formulae will be helpful to those who work in various applied areas for the computations required in their problems.

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Bibliography

  1. Alagar V.S.,The distribution of the distance between random points, J. Appl. Prob.,13 (1976), 558–566.

    Article  MathSciNet  MATH  Google Scholar 

  2. Borel E.,Principes et Formules Classiques du Calcul des Probabilités, Gauthier-Villars, Paris, (1925).

    MATH  Google Scholar 

  3. Christofides N., Eilon, S.,Expected distances in distribution problems, Opl. Res. Q.,20, (1969), 437–443.

    Google Scholar 

  4. Coleman R.,Random paths through convex bodies, J. Appl. Prob.,6 (1969), 430–441.

    Article  MATH  Google Scholar 

  5. Coleman R.,Random paths through rectangles and cubes. Metallography,6 (1973), 103–114.

    Article  Google Scholar 

  6. Crofton M. W.,Geometrical theorems relating to mean values, Proc. London Math. Soc.,8 (1877), 304–309.

    Article  Google Scholar 

  7. Crofton M. W.,Probability, in Encyclopaedia Britannica, 9th ed., Vol.19 (1885), pp. 768–788.

  8. Daley D.J.,Solution to problem 75–12. An average distance, SIAM Rev.,18 (1976), 498–499.

    Article  Google Scholar 

  9. Eilon S., Watson-Gandy C.D.T., Christofides N.,Distribution Management: Mathematical Modelling and Practical Analysis, Griffin, London, (1971).

    Google Scholar 

  10. Fairthorne D.,Distances between pairs of points in towns of simple geometrical shapes, Proc. Second International Symposium on the Theory of Road Traffic Flow, OECD, Paris, (1965), pp. 391–406.

  11. Gaboune B., Laporte G., Soumis F.,Expected distances between two uniformly distributed random points in rectangles and rectangular parallelepipeds, J. Opl. Res. Soc.,44(5) (1993), 513–519.

    Article  MATH  Google Scholar 

  12. Ghosh B.,On the distribution of random distances in a rectangle, Science and Culture,8(9) (1943a), 388.

    MathSciNet  Google Scholar 

  13. Ghosh B.,On random distances between two rectangles, Science and Culture,8(11) (1943b), 464.

    MathSciNet  Google Scholar 

  14. Ghosh B.,Topographie variation in statistical field, Calcutta Statist Assoc. Bull.,2(5) (1949), 11–28.

    Google Scholar 

  15. Ghosh B.,Random distances within a rectangle and between two rectangles, Bull. Calcutta Math. Soc.,43 (1951), 17–24.

    MathSciNet  MATH  Google Scholar 

  16. Horowitz, M.,Probability of random paths across elementary geometrical shapes, J. Appl. Prob.,2 (1965), 169–177.

    Article  MathSciNet  Google Scholar 

  17. Hsu A.C.,Expected distance between two random points in a polygon, M.Sc Dissertation, Department of Civil Engineering, MIT, (1990).

  18. Kendall M.G., Moran P.A.P.,Geometrical Probability, Griffin, London, (1963).

    MATH  Google Scholar 

  19. Kuchel, P.W., Vaughan R.J.,Average lengths of chords in a sguare, Math. Mag.,54(5) (1981), 261–269.

    Article  MathSciNet  MATH  Google Scholar 

  20. Larson R.C., Odoni A.R.,Urban Operations Research, Prentice-Hall, Englewood Cliffs, N.J., (1981).

    Google Scholar 

  21. Marsaglia G., Narasimhan B.G., Zaman A.,The distance between random points in rectangles, Commun. Statist.—Theory Meth.,19(11) (1990), 4199–4212.

    Article  MathSciNet  MATH  Google Scholar 

  22. Oser H.J.,Problem 75-12. An average distance, SIAM Rev.,18 (1976), 497.

    Article  Google Scholar 

  23. Ruben H.,On the distance between points in polygons, in Geometrical Probability and Biological Structures: Buffon's 200th Anniversary, R.E. Miles and J. Serra eds. Lecture Notes in Biomathematics,23 (1978), Springer-Verlag, Berlin pp. 49–69.

    Google Scholar 

  24. Sheng T.K.,The distance between two random points in plane regions, Adv. Appl. Prob.,17 (1985), 748–773.

    Article  MATH  Google Scholar 

  25. Solomon H.,Geometric Probability, SIAM. (1978).

  26. Stone R.E.,Some average distance results, Transp. Sci.,25 (1991), 83–91.

    Google Scholar 

  27. Vaughan R.J.,Solution to problem 75-12. An average distance, SIAM Rev.,18 (1976), 500.

    Google Scholar 

  28. Vaughan R.J.,Approximate formulas for the average distances associated with zones, Transp. Sci.,18 (1984), 231–244.

    Article  Google Scholar 

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The third Author has partially been supported by C.N.R..

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Mathai, A.M., Moschopoulos, P. & Pederzoli, G. Random points associated with rectangles. Rend. Circ. Mat. Palermo 48, 163–190 (1999). https://doi.org/10.1007/BF02844387

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  • DOI: https://doi.org/10.1007/BF02844387

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