5.3. (Sec. 5.2 2) Let

where U \> ••. , II N are N numbers and x\> ... , X N are independent, each with the distribution N(O, l:). Prove that the distribution of R2/O - R2) is independent of U I, ... , UN' [Hint: There is an orthogonal N X N matrix C that carries (Ill"'" LIN) into a vector proportional to (1/ {N, ... , 1/ {N).]

R . () 1-R2 Cu-Eux (xx') 'ux