A Table for Computing Trivariate Normal Probabilities

For c~nvenience in the folllowing discussion let X, P, ?r and .Z be random variables with a trfvariate normal.distribution such that p ) denote the probability that X <, h, Y <, k, Z zm, let C(h.k. m; p12.P13.23 P ) denoie the probability that X 2 b, Y L k, and let D(h.k.P12.P13.23 Z m.Several tables have been prepared from which certain particular values of the trivariate normal fntcgral can be obtained.A tabulation of the area of hyperspherical simpP f ces k s given by H. Ruben [ 1 ] .The func -.tion Ruben has tabulated as Gn(x) is, for the case n = 3. equal to G(O, 0, 0 ; l / x , 1/x, l / x ) and the.tabulation is for x 2 ( l ) l l .This probability can be computed direc91:/, however, as a special case of the wellknown formula (for example, nee [ 2) ), Short tabulations of @(laDh, h; 112, 112, 1/21 have been pzablhhed by Do Telchroew ( 31 for l a 6 = a(.0116.09 and by Po No SomesviMe [ B 1 1 for $ = Q(r,8)20,513, In addfiaiow 60 these pubBEshed tables, &!ere are some urnpublished tables [ 41 giving C(h, b, h; d, p0 pB for p = 1/(1 + 6 1 awd 1/4, h 5 8136,518 and for p 6 O(I, 990,9, 69 e O(I, 218" Methods b r computing D(h, k,, m: q2" p 3o p 231 have been given by M, Q, Menddl I 51 , R, L, Piaekett [ 61 , anal S, 6, Das f 71 .The method of Mendall ia to expreso the t.rPvrarPaQe normal density as %he ihmvesee of The fundamental formulas for Case Qbb: hZ.0, k ZOO m 2 0 or h z 0 8 k ZOO, ni 2 0 , and, The Sfunction is tabulated for 0 < b 5 l o but it is possible to obtdn values for 1 4 b < '= by use of one of the blbwing formulas, a ?0, b .0: s S( hab, l i b , l/a), . ., If a > 1, b .% t h e n t2:48 should be usid, and if' O e a 5 P o b 1 Yaen r92.5) * .should be used.Values for negative h. a, or b may be obtained by using ' Note that (2,4) and 62. 5) requbbe bdth a and b to be positive and hence when ;a or b is negative (2.79 or $2.8) should be applied before $2.4) or 62. 5).Othsr ,useful formulas are: normal po,aalaticans," BQsmetrika, Voll, 48 (lL954), pao 200-22%.F.