6. Let XI"' " x" be independently normally distributed with common variance a 2 and means ~., ... , ~". and let Z, = E'j=. aij be an orthogonal transformation (that is, E;'=. aijaik = 1 or 0 as j = k or j ,;, k) . The Z's are normally distributed with common variance a2 and means tj = Ea ii~j' [The density of the Z 's is obtained from that of the X's by substituting Xi = Ebjjzj , where (bi) is the inverse of the matrix (ai ;). and multiplying by the Jacobian, which is 1.]