Let X1,..., Xn be independently normally distributed with common variance 2 and means 1,..., n, and let

Let X1,..., Xn be independently normally distributed with common variance σ2 and means ζ1,..., ζn, and let Zi = n j=1 ai j X j be an orthogonal transformation (that is, n i=1 ai jaik = 1 or 0 as j = k or j = k). The Z’s are normally distributed with common variance σ2 and means ζi = ai j ξ j .

[The density of the Z’s is obtained from that of the X’s by substituting xi =

bi jzj , where (bi j) is the inverse of the matrix (ai j), and multiplying by the Jacobian, which is 1.]