3.13. (Sec. 3.3) Let Xa be distributed according to N( 'Yca, l:), a = 1, ... , N, where r.c~ > O. Show that the distribution of g = (l/r.c~)r.caXa is N[ 'Y,(l/r.c~)l:].

Show that E = r.a(Xa - gcaXXa - gca)' is independently distributed as r.~.::l ZaZ~, where ZI'"'' ZN are independent, each with distribution N(O, l:).

[Hint: Let Za = r.bo/lX/l' where bN/l = C/l/,;r;'I and B is orthogonal.]