66. (i) Let (XI' YI ) , ... ,(Xn , y,.) be a sample from the bivaria~ normal_distribution (74), and let Sf = '£(X; - X)2, Sl2 = '£(X; - X)(Y; - Y), sf = '£(Y, - Y)2. Then (s], Sl2' Sf) are independently distributed of (X, Y), and their joint distribution is the same as that of (f.7:lx:2, '£7:lX;'y,', '£7:ly,'2), where (X;', Y,'), i = 1, . . . , n - I, are a sample from the distribution (74) with = '1/ = O. (ii) Let Xl' . . . ' Xnr and YI , ••• , Ynr be two samples from N(O,1) . Then the joint density of Sr = '£X;2, Sl2 = '£X;Y" Sf = '£y,2 is 1 I __( 2 2 _ 2 ),(nr -3l [ .L( 2 2)] 4'ITr(m _ 1) SlS2 S12 exp - 2 SI + S2 for Sf2 s srsi, and zero elsewhere.

(iii) The joint density of the statistics (sf, S12 ' Si) of part (i) is (85) ( 2 2 2 i ( n - 4) SIS2 - S12) [1 (Sf 2pS12 s?)] lexp- ----+- 4wr(n-2)(oTVI- p2)"- 2(I- p2) 0 2 ar T 2 for S[z s sfs?, and zero elsewhere. [(i): Make an orthogonal transformation from XI"' " X; to X{,. . . , X~ such that x,; = {n X, and apply the same orthogonal transformation also to Y1. · · · , y" . Then y,: ,,; {nY, n-l n L X;Y,' = L (X, - X)(y, - Y), i-I i-I n-l n L x,/2 = L (x, - x)2, i-I i-I n-l n L y,/2 = L (y, - Y( i-I i-I The pairs of variables (X{ , Y\), .. . ,( X~ Y:) are independent, each with a bivariate normal distribution with the same variances and correlation as those of (X, Y) and with means E(Xf) = E(Y,/) = ° for i = 1, ... , n - 1. (ii): Consider first the joint distribution of S12 = LX;Y, and si = Ly,2 given XI" ' " Xm • Letting ZI = S12/ VLX; and making an orthogonal transformation from Y1, • •• , Ym to ZI" . . , Zm so that si = L:"_I Z;, the variables ZI and L7~2 Z; = si- Zf are independently distributed as N(O,I) and X~ -I respectively. From this the joint conditional density of S12 = Sl ZI and si is obtained by a simple transformation of variables. Since the conditional distribution depends on the x's only through sf, the joint density of Sf, S12'si is found by multiplying the above conditional density by the marginal one of S~. which is X;,. The proof is completed through use of the identity r[!(m-l)]r(!m)= (;r(m-l) 2m - 2 (iii): If (X', Y/) = (X{, Y{;. . . Y:,) is a sample from a bivariate normal distribution with = 7J = 0, then T = (L X;2 ,LX;Y,',L y,/2) is sufficient for (J = (0, p, T), and the density of T is obtained from that given in part (ii) for (Ju = (1,0,1) through the identity [Chapter 3, Problem 14 (i)] T x: Y'( P9(t) = iIo(t) P9 ' x', y') x: Y'( . P90 ' x'; y') The result now follows from part (i) with m = n - 1.]