54 RANDOM NUMBERS 3.3.1 normal distribution (cf. exercise (Make a web site)
54 RANDOM NUMBERS 3.3.1 normal distribution (cf. exercise 1.2.10-16); therefore points in a differential volume dz 2.. . dzk of S occur with probability approximately proportional to exp (-(24 + . . . + $)/2). (It is at this point in the derivation that the chi- square method becomes only an approximation for large n.) The probability that V 2 v is now S~zl,…,zrc)inSandzl+…+z,,Iv exP(-(z? + …+Z:)/2)dz~…dZk exp (-(zy + . + . + zg)/2) dz2 . . . dzk . (20) J(zl,…,zk)in S Since the hyperplane (19) passes through the origin of k-dimensional space, the numerator in (20) is an integration over the interior of a (k -1)-dimensional hypersphere centered at the origin. An appropriate transformation to generalized polar coordinates with radius x and angles ~1, . . . , wk-2 transforms (20) into Sp