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The volume of an isotropic random parallelotope

  • Harold Ruben (a1)

Abstract

The p-content of the p-parallelotope ∇ p, n determined by p independent isotropic random points z 1, …, zp in ℝ n (1 < pn) can be expressed as a product of independent variates in two ways, by successive orthogonal projection onto linear subspaces and by radial projection of the points, enabling calculation of the actual distribution as well as the moments of ∇ p, n . This is done explicitly in several cases. The results also have interest in multivariate statistics.

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Postal address: Department of Mathematics, McGill University, 805 Sherbrooke St. W., Montreal, P.Q. Canada, H3A 2K6.

References

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[1] Anderson, T. W. (1958) An Introduction to Multivariate Statistical Analysis. Wiley, New York.
[2] Johnson, N. L. and Kotz, S. (1972) Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York.
[3] Kelker, D. (1970) Distribution theory of spherical distributions and a location-scale parameter generalization. Sankhya A 32, 419430.
[4] Kendall, M. G. and Stuart, A. (1960) The Advanced Theory of Statistics, Vol. 2. Hafner Press, New York.
[5] Miles, R. E. (1971) Isotropic random simplices. Adv. Appl. Prob. 3, 353382.
[6] Sommerville, D. M. Y. (1929) An Introduction to the Geometry of N Dimensions. Methuen, London.

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