ABSTRACT
G T Tllc usual techniques ava,ilablc in the litcraturc in cleriving integer morrierits arid tiistxib~~tioris of raudom volurnes of randoir~ geometrical cor~figuratioris are thc iritegral geometry tcchniqiics. The ;tuthoi. has earlier iritroduced a method based or1 fiinctio~ls of matrix argument arid Jacobi;tris of rr~atrix transformations and derived itrhitrary ~nornclits, not just integer rrlorneiits, and the exact tlisl,ril)utioris of the volume contents of raridorri 11-parallelotopcs a d p-sirnplices in Euclitienn n-spacc, p < 7 1 + l , for tkic follomirig situations: (1) all Ihe points arc eithcr irrsitit: a hypcrsphcre of iiliil. i.a(iius or have general distrihtions associated wit,h them such ;is gcnc.ralizctl type-l beta, type-? het:% or Gaussiim, (2) sorne 1)oints arc uriifolxily tlistrihut,txl over the s~lrfitce of a hypcrspkierc of radius 1 itllti the reniainirlg have generid distrihtions as in (1). In all thcsc derivations 110 result fimn integral gcorilctry is used. In the present paper the case wheii the points have some general matrix-v;ariate distrilhtions ;ts well as the case when some points arc rest,ricted to the s~lrfitce of a hypcrspherc ;trid the rerrlainirig poi~rts have general classes of tlistrihutioris as in ( l) will be consitlcrctl. Thc exact arbitrary riiornerits a.rid the exact tlistributions ol the vo l~me conlent ol a. rantloin parallelotope will be derived without using any result horn iritegral geometry. The densities of rnntloni tlistanccs~ areas and volurries in several particular cases will be writteri i l l terms of Bessel, Whittaker, and Gauss' hypergeometric fimcl;ions.
